INTERNATIONAL GONGRESS 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z">(n,k,L). Using these ideas, we can get the following.
Proposition 4.4. Let n = 1 , 2 , 3 , n = 1 , 2 , 3 , … n=1,2,3,dotsn=1,2,3, \ldotsn=1,2,3,…. The following are equivalent over R C A 0 R C A 0 RCA_(0)\mathrm{RCA}_{0}RCA0 :
  1. m G P H k n ( k = 2 , 3 , 4 , , m = 1 , 2 , 3 , ) m G P H k n ( k = 2 , 3 , 4 , … , m = 1 , 2 , 3 , … ) mGPH_(k)^(n)(k=2,3,4,dots,m=1,2,3,dots)m \mathrm{GPH}_{k}^{n}(k=2,3,4, \ldots, m=1,2,3, \ldots)mGPHkn(k=2,3,4,…,m=1,2,3,…).
  2. G P H 2 n G P H 2 n GPH_(2)^(n)\mathrm{GPH}_{2}^{n}GPH2n on N N N\mathbb{N}N : for any largeness notion L L L\mathbb{L}L, there exists a finite set F N F ⊆ N F subeNF \subseteq \mathbb{N}F⊆N such that F F FFF is 1 -dense ( n , 2 , L ) ( n , 2 , L ) (n,2,L)(n, 2, \mathbb{L})(n,2,L).
To give a characterization of GPH, we consider the following variants of the infinite Ramsey theorem which was originally introduced by Flood [11].
Definition 4.5 (Ramsey-type weak Kônig's lemma). An infinite homogeneous function for an infinite ( n , k ) ( n , k ) (n,k)(n, k)(n,k)-coloring tree T T TTT on X 0 X 0 X_(0)X_{0}X0 is a function h : [ X ] n k h : [ X ] n → k h:[X]^(n)rarr kh:[X]^{n} \rightarrow kh:[X]n→k such that X X 0 X ⊆ X 0 X subeX_(0)X \subseteq X_{0}X⊆X0 is infinite and for any m N m ∈ N m inNm \in \mathbb{N}m∈N, there exists p T p ∈ T p in Tp \in Tp∈T such that h [ X m ] n = p [ X m ] n h ↾ [ X ∩ m ] n = p ↾ [ X ∩ m ] n h↾[X nn m]^(n)=p↾[X nn m]^(n)h \upharpoonright[X \cap m]^{n}=p \upharpoonright[X \cap m]^{n}h↾[X∩m]n=p↾[X∩m]n.
We define two forms of the Ramsey-type weak Kónig's lemma, R W K L k n R W K L k n RWKL_(k)^(n)\mathrm{RWKL}_{k}^{n}RWKLkn and R W K L k n R W K L k n − RWKL_(k)^(n-)\mathrm{RWKL}_{k}^{n-}RWKLkn−, as follows:
  • R W K L k n R W K L k n − RWKL_(k)^(n-)\mathrm{RWKL}_{k}^{n-}RWKLkn− : for any infinite ( n , k ) ( n , k ) (n,k)(n, k)(n,k)-coloring tree T T TTT on N N N\mathbb{N}N, there exists an infinite homogeneous function for T T TTT ( n 1 n ≥ 1 n >= 1n \geq 1n≥1 and k 2 k ≥ 2 k >= 2k \geq 2k≥2 ),
  • R W K L n k R W K L k n , R W K L n R W K L n R W K L ∞ n − ≡ ∀ k R W K L k n − , R W K L ∞ ∞ − ≡ ∀ n R W K L ∞ n − RWKL_(oo)^(n-)-=AA kRWKL_(k)^(n-),RWKL_(oo)^(oo-)-=AA nRWKL_(oo)^(n-)\mathrm{RWKL}_{\infty}^{n-} \equiv \forall k \mathrm{RWKL}_{k}^{n-}, \mathrm{RWKL}_{\infty}^{\infty-} \equiv \forall n \mathrm{RWKL}_{\infty}^{n-}RWKL∞n−≡∀kRWKLkn−,RWKL∞∞−≡∀nRWKL∞n−,
  • R W K L k n R W K L k n RWKL_(k)^(n)\mathrm{RWKL}_{k}^{n}RWKLkn : for any infinite ( n , k ) ( n , k ) (n,k)(n, k)(n,k)-coloring tree T T TTT on N N N\mathbb{N}N, there exists a constant infinite homogeneous function for T T TTT ( n 1 n ≥ 1 n >= 1n \geq 1n≥1 and k 2 k ≥ 2 k >= 2k \geq 2k≥2 ),
  • R W K L n k R W K L k n , R W K L n R W K L n R W K L ∞ n ≡ ∀ k R W K L k n , R W K L ∞ ∞ ≡ ∀ n R W K L ∞ n RWKL_(oo)^(n)-=AA kRWKL_(k)^(n),RWKL_(oo)^(oo)-=AA nRWKL_(oo)^(n)\mathrm{RWKL}_{\infty}^{n} \equiv \forall k \mathrm{RWKL}_{k}^{n}, \mathrm{RWKL}_{\infty}^{\infty} \equiv \forall n \mathrm{RWKL}_{\infty}^{n}RWKL∞n≡∀kRWKLkn,RWKL∞∞≡∀nRWKL∞n.
Note that the original definition of Ramsey-type weak Kőnig's lemma by Flood is our RWKL 2 1 2 1 _(2)^(1){ }_{2}^{1}21. 7 7 ^(7){ }^{7}7 Over RCA 0 0 _(0)_{0}0, it is strictly in-between WKL and DNR (see [11,12]). Variants of Ramsey-type weak Kónig's lemma with homogeneous functions are introduced and studied by Bienvenu, Patey, and Shafer in [2] and the definition of R W K L k n R W K L k n − RWKL_(k)^(n-)\mathrm{RWKL}_{k}^{n-}RWKLkn− is inspired by them.
Theorem 4.5. Let n 1 n ≥ 1 n >= 1n \geq 1n≥1 or n = n = ∞ n=oon=\inftyn=∞ and k 2 k ≥ 2 k >= 2k \geq 2k≥2 or k = k = ∞ k=ook=\inftyk=∞. The following are equivalent over R C A 0 R C A 0 RCA_(0)\mathrm{RCA}_{0}RCA0 :
  1. G P H k n G P H k n GPH_(k)^(n)\mathrm{GPH}_{k}^{n}GPHkn.
  2. FIRT k n FIRT k n FIRT_(k)^(n)\operatorname{FIRT}_{k}^{n}FIRTkn.
  3. RWKL k n k n _(k)^(n){ }_{k}^{n}kn.
  4. R T k n + R W K L k n R T k n + R W K L k n − RT_(k)^(n)+RWKL_(k)^(n-)\mathrm{RT}_{k}^{n}+\mathrm{RWKL}_{k}^{n-}RTkn+RWKLkn−.
Proof. It is enough to show the equivalence for the case n 1 n ≥ 1 n >= 1n \geq 1n≥1 and k 2 k ≥ 2 k >= 2k \geq 2k≥2. Equivalence 3 4 3 ↔ 4 3harr43 \leftrightarrow 43↔4 is easy from the definition. If f : [ N ] < N N f : [ N ] < N → N f:[N]^( < N)rarrNf:[\mathbb{N}]^{<\mathbb{N}} \rightarrow \mathbb{N}f:[N]<N→N is asymptotically stable, then L = { F : G L = { F : ∃ G ⊆ L={F:EE G sube\mathbb{L}=\{F: \exists G \subseteqL={F:∃G⊆
7 The original name in [11] was "Ramsey-type Kónig's lemma", but "Ramsey-type weak Kónig's lemma" turned to be the standard name in the later works.
F | G | > f ( G ) } F | G | > f ( G ) } F|G| > f(G)}F|G|>f(G)\}F|G|>f(G)} is a largeness notion, which implies 1 2 1 → 2 1rarr21 \rightarrow 21→2. Conversely, if L L L\mathbb{L}L is a largeness notion, then a function f f fff defined as f ( F ) = min { | G | 1 : G F G L } { | F | } f ( F ) = min { | G | − 1 : G ⊆ F ∧ G ∈ L } ∪ { | F | } f(F)=min{|G|-1:G sube F^^G inL}uu{|F|}f(F)=\min \{|G|-1: G \subseteq F \wedge G \in \mathbb{L}\} \cup\{|F|\}f(F)=min{|G|−1:G⊆F∧G∈L}∪{|F|} is asymptotically stable and F L | F | > f ( F ) F ∈ L ↔ | F | > f ( F ) F inLharr|F| > f(F)F \in \mathbb{L} \leftrightarrow|F|>f(F)F∈L↔|F|>f(F). This implies 2 1 2 → 1 2rarr12 \rightarrow 12→1. Implication 3 1 3 → 1 3rarr13 \rightarrow 13→1 is a standard compactness argument which we have seen in Proposition 3.1. To show 1 3 1 → 3 1rarr31 \rightarrow 31→3, let T T TTT be an infinite ( n , k ) ( n , k ) (n,k)(n, k)(n,k)-coloring tree on N N N\mathbb{N}N with no infinite constant homogeneous function. Define L L L\mathbb{L}L as F L F ∈ L F inLF \in \mathbb{L}F∈L if there is no p T p ∈ T p in Tp \in Tp∈T such that p p ppp is constant on [ F ] n [ F ] n [F]^(n)[F]^{n}[F]n. Then, one can check that L L L\mathbb{L}L is a largeness notion, and hence by 1 , there exists a finite set F 0 N F 0 ⊆ N F_(0)subeNF_{0} \subseteq \mathbb{N}F0⊆N which is 1 -dense ( n , k , L ) ( n , k , L ) (n,k,L)(n, k, \mathbb{L})(n,k,L). Take some p T p ∈ T p in Tp \in Tp∈T so that dom ( p ) [ F 0 ] n dom ⁡ ( p ) ⊇ F 0 n dom(p)supe[F_(0)]^(n)\operatorname{dom}(p) \supseteq\left[F_{0}\right]^{n}dom⁡(p)⊇[F0]n, then there must exist H F 0 H ⊆ F 0 H subeF_(0)H \subseteq F_{0}H⊆F0 such that H L H ∈ L H inLH \in \mathbb{L}H∈L and p p ppp is constant on [ H ] n [ H ] n [H]^(n)[H]^{n}[H]n, which is a contradiction.
In case n = 3 , 4 , 5 , n = 3 , 4 , 5 , … n=3,4,5,dotsn=3,4,5, \ldotsn=3,4,5,…, any of the statements in the above theorem is just equivalent to A C A 0 A C A 0 ACA_(0)\mathrm{ACA}_{0}ACA0, so we mostly interested in the case n = 1 n = 1 n=1n=1n=1 and 2 . On the other hand, unlike R T 2 1 R T 2 1 RT_(2)^(1)\mathrm{RT}_{2}^{1}RT21 or P H 2 1 P H 2 1 PH_(2)^(1)\mathrm{PH}_{2}^{1}PH21, the principle G P H 2 1 G P H 2 1 GPH_(2)^(1)\mathrm{GPH}_{2}^{1}GPH21 is still not trivial since R W K L 2 1 R W K L 2 1 RWKL_(2)^(1)\mathrm{RWKL}_{2}^{1}RWKL21 (which is equivalent to R W K L 2 1 R W K L 2 1 − RWKL_(2)^(1-)\mathrm{RWKL}_{2}^{1-}RWKL21− ) is not provable within R C A 0 R C A 0 RCA_(0)\mathrm{RCA}_{0}RCA0. This may be interpreted as saying that the generalized version of the Paris-Harrington principle cannot be proved without using some compactness argument. In general, R W K L k n R W K L k n − RWKL_(k)^(n-)\mathrm{RWKL}_{k}^{n-}RWKLkn− is easily implied by W K L 0 W K L 0 WKL_(0)\mathrm{WKL}_{0}WKL0, but we do not know whether it is strictly weaker than WKL over R C A 0 R C A 0 RCA_(0)\mathrm{RCA}_{0}RCA0 or not in case n 2 n ≥ 2 n >= 2n \geq 2n≥2.

4.3. Iterations of generalized P H P H PH\mathbf{P H}PH and correctness statements

The iterated version of GPH can be related to stronger correctness statements.
Theorem 4.6. Let k = 2 k = 2 k=2k=2k=2 or k = k = ∞ k=ook=\inftyk=∞. Then I t G P H k 2 I t G P H k 2 ItGPH_(k)^(2)\mathrm{ItGPH}_{k}^{2}ItGPHk2 is equivalent to r Π 2 1 r Π 2 1 rPi_(2)^(1)\mathrm{r} \Pi_{2}^{1}rΠ21-corr ( W K L 0 + R T k 2 ) W K L 0 + R T k 2 (WKL_(0)+RT_(k)^(2))\left(\mathrm{WKL}_{0}+\mathrm{RT}_{k}^{2}\right)(WKL0+RTk2) over W K L 0 W K L 0 WKL_(0)\mathrm{WKL}_{0}WKL0.
Over A C A 0 A C A 0 ′ ACA_(0)^(')\mathrm{ACA}_{0}^{\prime}ACA0′, any Π 2 1 Π 2 1 Pi_(2)^(1)\Pi_{2}^{1}Π21-formula (of possibly nonstandard length) is equivalent to a
r Π 2 1 r Π 2 1 rPi_(2)^(1)\mathrm{r} \Pi_{2}^{1}rΠ21-formula, and thus r Π 2 1 r Π 2 1 rPi_(2)^(1)\mathrm{r} \Pi_{2}^{1}rΠ21-truth predicate is actually the truth predicate for all Π 2 1 Π 2 1 Pi_(2)^(1)\Pi_{2}^{1}Π21-formulas. Furthermore, It is known that r Π 2 1 corr ( A C A 0 ) r Π 2 1 − corr ⁡ A C A 0 rPi_(2)^(1)-corr(ACA_(0))\mathrm{r} \Pi_{2}^{1}-\operatorname{corr}\left(\mathrm{ACA}_{0}\right)rΠ21−corr⁡(ACA0) is equivalent to A C A 0 . 8 A C A 0 ′ . 8 ACA_(0)^(').^(8)\mathrm{ACA}_{0}^{\prime} .{ }^{8}ACA0′.8 So we simply write Π 2 1 corr ( T ) Π 2 1 − corr ⁡ ( T ) Pi_(2)^(1)-corr(T)\Pi_{2}^{1}-\operatorname{corr}(T)Π21−corr⁡(T) for r Π 2 1 corr ( T ) r Π 2 1 − corr ⁡ ( T ) rPi_(2)^(1)-corr(T)\mathrm{r} \Pi_{2}^{1}-\operatorname{corr}(T)rΠ21−corr⁡(T) if T A C A 0 T ⊇ A C A 0 T supeACA_(0)T \supseteq \mathrm{ACA}_{0}T⊇ACA0.
Theorem 4.7. The following are equivalent over R C A 0 R C A 0 RCA_(0)\mathrm{RCA}_{0}RCA0 :
  1. RT.
  2. GPH.
  3. ItGPH k n ( n = 3 , 4 , , k = 2 , 3 , , ) ItGPH k n ⁡ ( n = 3 , 4 , … , k = 2 , 3 , … , ∞ ) ItGPH_(k)^(n)(n=3,4,dots,k=2,3,dots,oo)\operatorname{ItGPH}_{k}^{n}(n=3,4, \ldots, k=2,3, \ldots, \infty)ItGPHkn⁡(n=3,4,…,k=2,3,…,∞).
  4. Π 2 1 corr ( A C A 0 ) Π 2 1 − corr ⁡ A C A 0 Pi_(2)^(1)-corr(ACA_(0))\Pi_{2}^{1}-\operatorname{corr}\left(\mathrm{ACA}_{0}\right)Π21−corr⁡(ACA0).
Theorem 4.8. Over R C A 0 R C A 0 RCA_(0)\mathrm{RCA}_{0}RCA0, ItGPH is equivalent to Π 2 1 Π 2 1 Pi_(2)^(1)\Pi_{2}^{1}Π21 - corr ( A C A 0 ) corr ⁡ A C A 0 ′ corr(ACA_(0)^('))\operatorname{corr}\left(\mathrm{ACA}_{0}^{\prime}\right)corr⁡(ACA0′).
We will see the proofs of these theorems using indicators in the next section.
The strength of I t G P H 2 2 I t G P H 2 2 ItGPH_(2)^(2)\mathrm{ItGPH}_{2}^{2}ItGPH22 or I t G P H 2 I t G P H 2 ItGPH^(2)\mathrm{ItGPH}^{2}ItGPH2 is rather unclear. It is not difficult to check that R C A 0 + I t G P H 2 2 R C A 0 + I t G P H 2 2 RCA_(0)+ItGPH_(2)^(2)\mathrm{RCA}_{0}+\mathrm{ItGPH}_{2}^{2}RCA0+ItGPH22 implies R T 2 R T 2 RT^(2)\mathrm{RT}^{2}RT2 and W K L 0 + R T 2 2 + I Σ 1 1 W K L 0 + R T 2 2 + I Σ 1 1 WKL_(0)+RT_(2)^(2)+ISigma_(1)^(1)\mathrm{WKL}_{0}+\mathrm{RT}_{2}^{2}+I \Sigma_{1}^{1}WKL0+RT22+IΣ11 implies ItGPH 2 2 ^(2){ }^{2}2 as in the proof of Proposition 3.1. (Note that even ItGPH does not imply Σ 1 1 ∣ Σ 1 1 ∣Sigma_(1)^(1)\mid \Sigma_{1}^{1}∣Σ11 since I Σ 1 1 I Σ 1 1 ISigma_(1)^(1)I \Sigma_{1}^{1}IΣ11 is never implied from
any true Π 2 1 Π 2 1 Pi_(2)^(1)\Pi_{2}^{1}Π21-statement.) In particular, they are true in any ω ω omega\omegaω-models of W K L 0 + R T 2 2 W K L 0 + R T 2 2 WKL_(0)+RT_(2)^(2)\mathrm{WKL}_{0}+\mathrm{RT}_{2}^{2}WKL0+RT22. Meanwhile, the following questions are still open.
Question 4.6. quad\quad 1. Is I t G P H 2 2 I t G P H 2 2 ItGPH_(2)^(2)\mathrm{ItGPH}_{2}^{2}ItGPH22 equivalent to R T 2 R T 2 RT^(2)\mathrm{RT}^{2}RT2 over W K L 0 W K L 0 WKL_(0)\mathrm{WKL}_{0}WKL0 ?
  1. Does A C A 0 A C A 0 ACA_(0)\mathrm{ACA}_{0}ACA0 imply ItGPH 2 2 ^(2){ }^{2}2 or I t G P H 2 2 I t G P H 2 2 ItGPH_(2)^(2)\mathrm{ItGPH}_{2}^{2}ItGPH22 ?

5. INDICATORS AND CORRECTNESS STATEMENTS

The notion of indicators is introduced by Kirby and Paris [23,32] to show several independence results from PA, and its theory is organized systematically by Kaye [21]. The argument of indicators can connect first-order objects with second-order objects by means of nonstandard models. Recently, indicators have been used to calibrate the proof-theoretic strength of the infinite Ramsey theorem in the context of reverse mathematics [ 6 , 24 , 34 , 46 ] [ 6 , 24 , 34 , 46 ] [6,24,34,46][6,24,34,46][6,24,34,46].

5.1. Models of first- and second-order arithmetic

To introduce the argument of indicators, we first set up basic model theory of firstand second-order arithmetic. For the details, see [ 16 , 21 , 26 , 39 ] [ 16 , 21 , 26 , 39 ] [16,21,26,39][16,21,26,39][16,21,26,39]. A structure for L 1 L 1 L_(1)\mathscr{L}_{1}L1 is a 6-tuple M = ( M ; 0 M , 1 M , + M , × M , M ) M = M ; 0 M , 1 M , + M , × M , ≤ M M=(M;0^(M),1^(M),+^(M),xx^(M), <= ^(M))M=\left(M ; 0^{M}, 1^{M},+{ }^{M}, \times{ }^{M}, \leq^{M}\right)M=(M;0M,1M,+M,×M,≤M). (We often omit the superscript M M MMM if it is clear from the context.) An L 1 L 1 L_(1)\mathscr{L}_{1}L1-structure = ( ; 0 , 1 , + , × , ) ℜ = ( ℜ ; 0 , 1 , + , × , ≤ ) ℜ=(ℜ;0,1,+,xx, <= )\Re=(\Re ; 0,1,+, \times, \leq)ℜ=(ℜ;0,1,+,×,≤) where 0 , 1 , + , × , 0 , 1 , + , × , ≤ 0,1,+,xx, <=0,1,+, \times, \leq0,1,+,×,≤ are usual is called the standard model, and an L 1 L 1 L_(1)\mathscr{L}_{1}L1-structure is said to be nonstandard if it is not isomorphic to ℜ ℜ\Reℜ. When we consider an expanded language L 1 U L 1 ∪ U → L_(1)uu vec(U)\mathscr{L}_{1} \cup \vec{U}L1∪U→ where U = U 1 , , U k U → = U 1 , … , U k vec(U)=U_(1),dots,U_(k)\vec{U}=U_{1}, \ldots, U_{k}U→=U1,…,Uk are secondorder constants, an L 1 U L 1 ∪ U → L_(1)uu vec(U)\mathscr{L}_{1} \cup \vec{U}L1∪U→-structure is a pair ( M , U M ) M , U → M (M, vec(U)^(M))\left(M, \vec{U}^{M}\right)(M,U→M) where M M MMM is an L 1 L 1 L_(1)\mathscr{L}_{1}L1-structure and U i M U i ⊆ M U_(i)sube MU_{i} \subseteq MUi⊆M. We may consider N N N\mathbb{N}N as a special second-order constant which satisfies x x N ∀ x x ∈ N AA xx inN\forall x x \in \mathbb{N}∀xx∈N, in other words, N M = M N M = M N^(M)=M\mathbb{N}^{M}=MNM=M for any M M MMM. For second-order arithmetic, we use Henkin semantics. A structure for L 2 L 2 L_(2)\mathscr{L}_{2}L2 is a pair ( M , S ) ( M , S ) (M,S)(M, S)(M,S) where M M MMM is an L 1 L 1 L_(1)\mathscr{L}_{1}L1-structure and S P ( M ) S ⊆ P ( M ) S subeP(M)S \subseteq \mathcal{P}(M)S⊆P(M). Thus, any L 1 U L 1 ∪ U → L_(1)uu vec(U)\mathscr{L}_{1} \cup \vec{U}L1∪U→-structure can be considered as an L 2 L 2 L_(2)\mathscr{L}_{2}L2-structure.
Let M M MMM be a nonstandard model of EFA U U → vec(U)\vec{U}U→. We write [ M ] < M [ M ] < M [M]^( < M)[M]^{<M}[M]<M for the set of all "finite sets in M M MMM " (also called M M MMM-finite sets), in other words, [ M ] < M = ( [ N ] < N ) M [ M ] < M = [ N ] < N M [M]^( < M)=([N]^( < N))^(M)[M]^{<M}=\left([\mathbb{N}]^{<\mathbb{N}}\right)^{M}[M]<M=([N]<N)M. A nonempty proper subset I M I ⊊ M I⊊MI \subsetneq MI⊊M is said to be a cut if a < b b I a < b ∧ b ∈ I a < b^^b in Ia<b \wedge b \in Ia<b∧b∈I implies a I a ∈ I a in Ia \in Ia∈I for any a , b M a , b ∈ M a,b in Ma, b \in Ma,b∈M (denoted by I e M I ⊆ e M Isube_(e)MI \subseteq_{e} MI⊆eM ) and a + 1 I a + 1 ∈ I a+1in Ia+1 \in Ia+1∈I for any a I a ∈ I a in Ia \in Ia∈I. If I I III is a cut and φ ( x ) φ ( x ) varphi(x)\varphi(x)φ(x) is a Σ 0 U Σ 0 U → Sigma_(0)^( vec(U))\Sigma_{0}^{\vec{U}}Σ0U→-formula such that M φ ( a ) M ⊨ φ ( a ) M|==varphi(a)M \models \varphi(a)M⊨φ(a) for any a I a ∈ I a in Ia \in Ia∈I (resp. a M I a ∈ M ∖ I a in M\\Ia \in M \backslash Ia∈M∖I ), then there exists a M I a ∈ M ∖ I a in M\\Ia \in M \backslash Ia∈M∖I (resp. a I a ∈ I a in Ia \in Ia∈I ) such that M φ ( a ) M ⊨ φ ( a ) M|==varphi(a)M \models \varphi(a)M⊨φ(a). This principle is called overspill (resp. underspill). A cut I e M I ⊆ e M Isube_(e)MI \subseteq_{e} MI⊆eM is said to be semiregular if for any F [ M ] < M F ∈ [ M ] < M F in[M]^( < M)F \in[M]^{<M}F∈[M]<M with | F | min F , F I | F | ≤ min F , F ∩ I |F| <= min F,F nn I|F| \leq \min F, F \cap I|F|≤minF,F∩I is bounded in I I III.
In our study, models of W K L 0 W K L 0 WKL_(0)\mathrm{WKL}_{0}WKL0 play central roles. Here are two important theorems.
Theorem 5.1 (Harrington, see Section IX. 2 of [39]).
  1. For any countable model ( M , S ) RCA 0 ( M , S ) ⊨ RCA 0 (M,S)|==RCA_(0)(M, S) \models \operatorname{RCA}_{0}(M,S)⊨RCA0, there exists S ¯ S S ¯ ⊇ S bar(S)supe S\bar{S} \supseteq SS¯⊇S such that ( M , S ¯ ) ( M , S ¯ ) ⊨ (M, bar(S))|==(M, \bar{S}) \models(M,S¯)⊨ WKL. 9 9 ^(9){ }^{9}9
  2. W K L 0 W K L 0 WKL_(0)\mathrm{WKL}_{0}WKL0 is Π 1 1 Π 1 1 Pi_(1)^(1)\Pi_{1}^{1}Π11-conservative over R C A 0 R C A 0 RCA_(0)\mathrm{RCA}_{0}RCA0.
9 Model ( M , S ) ( M , S ) (M,S)(M, S)(M,S) is said to be countable if both of M M MMM and S S SSS are countable.
Theorem 5.2 (see, e.g., Theorems 7.1.5 and 7.1.7 of [26]). Let M M MMM be a model of EFA and I e M I ⊊ e M I⊊_(e)MI \subsetneq_{e} MI⊊eM be a cut. Then, I I III is semiregular if and only if ( I , Cod ( M / I ) ) W K L 0 ( I , Cod ⁡ ( M / I ) ) ⊨ W K L 0 (I,Cod(M//I))|==WKL_(0)(I, \operatorname{Cod}(M / I)) \models \mathrm{WKL}_{0}(I,Cod⁡(M/I))⊨WKL0, where Cod ( M / I ) = { F I : F [ M ] < M } Cod ⁡ ( M / I ) = F ∩ I : F ∈ [ M ] < M Cod(M//I)={F nn I:F in[M]^( < M)}\operatorname{Cod}(M / I)=\left\{F \cap I: F \in[M]^{<M}\right\}Cod⁡(M/I)={F∩I:F∈[M]<M}.

5.2. Indicators

Now we give the definition of indicators. Here, we slightly arrange the definition in [21] so as to fit better with second-order arithmetic.
Definition 5.1 (Indicators). Let U = U 1 , , U k U → = U 1 , … , U k vec(U)=U_(1),dots,U_(k)\vec{U}=U_{1}, \ldots, U_{k}U→=U1,…,Uk be second-order constants, and let T T ⊇ T supeT \supseteqT⊇ EFA be an L 2 L 2 L_(2)\mathscr{L}_{2}L2-theory.
  1. Let M M MMM be a countable nonstandard model of EFA U U → vec(U)\vec{U}U→. A Σ 0 U Σ 0 U → Sigma_(0)^( vec(U))\Sigma_{0}^{\vec{U}}Σ0U→-definable function Y : [ M ] < M M Y : [ M ] < M → M Y:[M]^( < M)rarr MY:[M]^{<M} \rightarrow MY:[M]<M→M is said to be an indicator for T T TTT on M M MMM if for each F , F [ M ] < M , Y ( F ) max F , Y ( F ) Y ( F ) F , F ′ ∈ [ M ] < M , Y ( F ) ≤ max F , Y ( F ) ≤ Y F ′ F,F^(')in[M]^( < M),Y(F) <= max F,Y(F) <= Y(F^('))F, F^{\prime} \in[M]^{<M}, Y(F) \leq \max F, Y(F) \leq Y\left(F^{\prime}\right)F,F′∈[M]<M,Y(F)≤maxF,Y(F)≤Y(F′) if F F F ⊆ F ′ F subeF^(')F \subseteq F^{\prime}F⊆F′, and
(cut) Y ( F ) > m Y ( F ) > m Y(F) > mY(F)>mY(F)>m for any m N m ∈ N m inNm \in \mathfrak{N}m∈N if and only if there exists a cut I e M I ⊊ e M I⊊eMI \subsetneq e MI⊊eM and
S Cod ( M / I ) S ⊆ Cod ⁡ ( M / I ) S sube Cod(M//I)S \subseteq \operatorname{Cod}(M / I)S⊆Cod⁡(M/I) such that ( I , S ) T , U i M I S ( I , S ) ⊨ T , U i M ∩ I ∈ S (I,S)|==T,U_(i)^(M)nn I in S(I, S) \models T, U_{i}^{M} \cap I \in S(I,S)⊨T,UiM∩I∈S for each U i U U i ∈ U → U_(i)in vec(U)U_{i} \in \vec{U}Ui∈U→ and F I F ∩ I F nn IF \cap IF∩I is unbounded in I I III.
  1. A Σ 0 U Σ 0 U → Sigma_(0)^( vec(U))\Sigma_{0}^{\vec{U}}Σ0U→-formula Y ( F , m ) Y ( F , m ) Y(F,m)Y(F, m)Y(F,m) is said to be an indicator for T T TTT if for any countable nonstandard model M E F A V M ⊨ E F A V → M|==EFA^( vec(V))M \models \mathrm{EFA}^{\vec{V}}M⊨EFAV→ with U V ( U U → ⊆ V → ( U → vec(U)sube vec(V)( vec(U)\vec{U} \subseteq \vec{V}(\vec{U}U→⊆V→(U→ is a subtuple of V V → vec(V)\vec{V}V→ ), Y Y YYY defines an indicator for T T TTT on M M MMM.
For a given indicator Y Y YYY, we define two statements " Y m Y ≥ m Y >= mY \geq mY≥m " and " Y Y YYY int m ≥ m >= m\geq m≥m " as follows:
Y m X 0 ( X 0 is infinite F fin X 0 Y ( F ) m ) Y int m a b Y ( [ a , b ) N ) m Y ≥ m ≡ ∀ X 0 X 0  is infinite  → ∃ F ⊆ fin  X 0 Y ( F ) ≥ m Y int  ≥ m ≡ ∀ a ∃ b Y [ a , b ) N ≥ m {:[Y >= m-=AAX_(0)(X_(0)" is infinite "rarr EE Fsube_("fin ")X_(0)Y(F) >= m)],[Y^("int ") >= m-=AA a EE bY([a,b)_(N)) >= m]:}\begin{aligned} & Y \geq m \equiv \forall X_{0}\left(X_{0} \text { is infinite } \rightarrow \exists F \subseteq_{\text {fin }} X_{0} Y(F) \geq m\right) \\ & Y^{\text {int }} \geq m \equiv \forall a \exists b Y\left([a, b)_{\mathbb{N}}\right) \geq m \end{aligned}Y≥m≡∀X0(X0 is infinite →∃F⊆fin X0Y(F)≥m)Yint ≥m≡∀a∃bY([a,b)N)≥m
Note that Y m Y ≥ m Y >= mY \geq mY≥m is ar Π 1 1 Π 1 1 Pi_(1)^(1)\Pi_{1}^{1}Π11-statement while Y i n t m Y i n t ≥ m Y^(int) >= mY^{\mathrm{int}} \geq mYint≥m is a Π 2 Π 2 Pi_(2)\Pi_{2}Π2-statement.
Theorem 5.3. Define Σ 0 Σ 0 Sigma_(0)\Sigma_{0}Σ0-formulas Y P H n ( F , m ) , Y P H ( F , m ) Y P H n ( F , m ) , Y P H ( F , m ) Y_(PH^(n))(F,m),Y_(PH)(F,m)Y_{\mathrm{PH}^{n}}(F, m), Y_{\mathrm{PH}}(F, m)YPHn(F,m),YPH(F,m), and Y I t P H k n ( F , m ) Y I t P H k n ( F , m ) Y_(ItPH_(k)^(n))(F,m)Y_{\mathrm{ItPH}_{k}^{n}}(F, m)YItPHkn(F,m) as follows:
  • Y P H n ( F , m ) m = max { k max F : F Y P H n ( F , m ) ↔ m = max k ′ ≤ max F : F Y_(PH^(n))(F,m)harr m=max{k^(') <= max F:F:}Y_{\mathrm{PH}^{n}}(F, m) \leftrightarrow m=\max \left\{k^{\prime} \leq \max F: F\right.YPHn(F,m)↔m=max{k′≤maxF:F is 1-dense ( n , k ) } { 0 } ( n = 2 , 3 , ) n , k ′ ∪ { 0 } ( n = 2 , 3 , … ) {:(n,k^('))}uu{0}(n=2,3,dots)\left.\left(n, k^{\prime}\right)\right\} \cup\{0\}(n=2,3, \ldots)(n,k′)}∪{0}(n=2,3,…),
  • Y P H ( F , m ) m = max { n max F : F Y P H ( F , m ) ↔ m = max n ′ ≤ max F : F Y_(PH)(F,m)harr m=max{n^(') <= max F:F:}Y_{\mathrm{PH}}(F, m) \leftrightarrow m=\max \left\{n^{\prime} \leq \max F: F\right.YPH(F,m)↔m=max{n′≤maxF:F is 1-dense ( n , 2 ) } { 0 } n ′ , 2 ∪ { 0 } {:(n^('),2)}uu{0}\left.\left(n^{\prime}, 2\right)\right\} \cup\{0\}(n′,2)}∪{0},
  • Y I t P H k n ( F , m ) m = max { m max F : F Y I t P H k n ( F , m ) ↔ m = max m ′ ≤ max F : F Y_(ItPH_(k)^(n))(F,m)harr m=max{m^(') <= max F:F:}Y_{\mathrm{ItPH}_{k}^{n}}(F, m) \leftrightarrow m=\max \left\{m^{\prime} \leq \max F: F\right.YItPHkn(F,m)↔m=max{m′≤maxF:F is m m ′ m^(')m^{\prime}m′-dense ( n , k ) } { 0 } ( n = 2 , 3 , ( n , k ) ∪ { 0 } ( n = 2 , 3 , … {:(n,k)}uu{0}(n=2,3,dots\left.(n, k)\right\} \cup\{0\}(n=2,3, \ldots(n,k)}∪{0}(n=2,3,… or n = n = ∞ n=oon=\inftyn=∞ and k = 2 , 3 , 4 , k = 2 , 3 , 4 , … k=2,3,4,dotsk=2,3,4, \ldotsk=2,3,4,… or k = ) k = ∞ ) k=oo)k=\infty)k=∞).
Then, we have the following:
  1. Y P H n Y P H n Y_(PH^(n))Y_{\mathrm{PH}^{n}}YPHn is an indicator for R C A 0 + 1 Σ n 1 0 R C A 0 + 1 Σ n − 1 0 RCA_(0)+1Sigma_(n-1)^(0)\mathrm{RCA}_{0}+1 \Sigma_{n-1}^{0}RCA0+1Σn−10.
  2. Y P H Y P H Y_(PH)Y_{\mathrm{PH}}YPH is an indicator for A C A 0 A C A 0 ACA_(0)\mathrm{ACA}_{0}ACA0.
  3. Y I t P H k n Y I t P H k n Y_(ItPH_(k)^(n))Y_{\mathrm{ItPH}_{k}^{n}}YItPHkn is an indicator for W K L 0 + R T k n W K L 0 + R T k n WKL_(0)+RT_(k)^(n)\mathrm{WKL}_{0}+\mathrm{RT}_{k}^{n}WKL0+RTkn.
In addition, these facts are provable within W K L 0 W K L 0 WKL_(0)\mathrm{WKL}_{0}WKL0.
Proof. For statements 1 and 2, one can reformulate the discussions of [21, SECTIoN 14.3]. Statement 3 is essentially due to Paris [32, EXAMPLE 2] (see also [6, THEOREM 1] and [34, LEMMA 3.2]). We sketch the proof for statement 3 for the case n = 2 , 3 , n = 2 , 3 , … n=2,3,dotsn=2,3, \ldotsn=2,3,… and k = 2 , 3 , k = 2 , 3 , … k=2,3,dotsk=2,3, \ldotsk=2,3,…

lows from Proposition 3.1 and overspill. For the left-to-right direction, let M M MMM be a countable nonstandard model of EFA U U → vec(U)\vec{U}U→ and let F [ M ] < M F ∈ [ M ] < M F in[M]^( < M)F \in[M]^{<M}F∈[M]<M be m m mmm-dense ( n , k ) ( n , k ) (n,k)(n, k)(n,k) for any m m ∈ ℜ m inℜm \in \Rem∈ℜ. By overspill, take d M d ∈ M ∖ ℜ d in M\\ℜd \in M \backslash \Red∈M∖ℜ such that F F FFF is d d ddd-dense ( n , k ) ( n , k ) (n,k)(n, k)(n,k). We will construct a countable decreasing sequence of M M MMM-finite sets { F i } i F i i ∈ ℜ {F_(i)}_(i inℜ)\left\{F_{i}\right\}_{i \in \Re}{Fi}i∈ℜ such that F i F i F_(i)F_{i}Fi is ( d i ) ( d − i ) (d-i)(d-i)(d−i)-dense ( n , k ) ( n , k ) (n,k)(n, k)(n,k) and
(i) if E [ M ] < M E ∈ [ M ] < M E in[M]^( < M)E \in[M]^{<M}E∈[M]<M and | E | E | E | ≤ E |E| <= E|E| \leq E|E|≤E, then E [ min F i , max F i ) N = E ∩ min F i ,  max F i N = ∅ E nn[minF_(i)", "maxF_(i))_(N)=O/E \cap\left[\min F_{i} \text {, } \max F_{i}\right)_{\mathbb{N}}=\emptysetE∩[minFi, maxFi)N=∅ for some i i ∈ ℜ i inℜi \in \Rei∈ℜ,
(ii) if p [ M ] < M p ∈ [ M ] < M p in[M]^( < M)p \in[M]^{<M}p∈[M]<M and p : [ F ] n k p : [ F ] n → k p:[F]^(n)rarr kp:[F]^{n} \rightarrow kp:[F]n→k, then for some i , F i i ∈ ℜ , F i i inℜ,F_(i)i \in \Re, F_{i}i∈ℜ,Fi is p p ppp-homogeneous. Once such a sequence is constructed, put I = { a M : i ( a < min F i ) } I = a ∈ M : ∃ i ∈ ℜ a < min F i I={a in M:EE i inℜ(a < minF_(i))}I=\left\{a \in M: \exists i \in \Re\left(a<\min F_{i}\right)\right\}I={a∈M:∃i∈ℜ(a<minFi)}. Then, F i I F i ∩ I F_(i)nn IF_{i} \cap IFi∩I is unbounded in I I III and U i M I Cod ( M / I ) U i M ∩ I ∈ Cod ⁡ ( M / I ) U_(i)^(M)nn I in Cod(M//I)U_{i}^{M} \cap I \in \operatorname{Cod}(M / I)UiM∩I∈Cod⁡(M/I). By Theorem 5.2, ( I , Cod ( M / I ) ) W K L 0 ( I , Cod ⁡ ( M / I ) ) ⊨ W K L 0 (I,Cod(M//I))|==WKL_(0)(I, \operatorname{Cod}(M / I)) \models \mathrm{WKL}_{0}(I,Cod⁡(M/I))⊨WKL0 since I I III is a semiregular cut by (i), and (ii) implies ( I , Cod ( M / I ) ) R T k n ( I , Cod ⁡ ( M / I ) ) ⊨ R T k n (I,Cod(M//I))|==RT_(k)^(n)(I, \operatorname{Cod}(M / I)) \models \mathrm{RT}_{k}^{n}(I,Cod⁡(M/I))⊨RTkn.
Finally, we construct { F i } i F i i ∈ ℜ {F_(i)}_(i inℜ)\left\{F_{i}\right\}_{i \in \Re}{Fi}i∈ℜ. Since [ M ] < M [ M ] < M [M]^( < M)[M]^{<M}[M]<M is countable, it is enough to show:
(i) if E [ M ] < M , | E | min E E ∈ [ M ] < M , | E | ≤ min E E in[M]^( < M),|E| <= min EE \in[M]^{<M},|E| \leq \min EE∈[M]<M,|E|≤minE and F F FFF is + 1 ℓ + 1 ℓ+1\ell+1ℓ+1-dense ( n , k ) ( n , k ) (n,k)(n, k)(n,k) then there exists F F F ′ ⊆ F F^(')sube FF^{\prime} \subseteq FF′⊆F which is ℓ ℓ\ellℓ-dense ( n , k ) ( n , k ) (n,k)(n, k)(n,k) such that E [ min F i , max F i ) N = E ∩ min F i , max F i N = ∅ E nn[minF_(i),maxF_(i))_(N)=O/E \cap\left[\min F_{i}, \max F_{i}\right)_{\mathbb{N}}=\emptysetE∩[minFi,maxFi)N=∅,
(ii)' if p [ M ] < M , p : [ F ] n k p ∈ [ M ] < M , p : [ F ] n → k p in[M]^( < M),p:[F]^(n)rarr kp \in[M]^{<M}, p:[F]^{n} \rightarrow kp∈[M]<M,p:[F]n→k and F F FFF is + 1 ℓ + 1 ℓ+1\ell+1ℓ+1-dense ( n , k ) ( n , k ) (n,k)(n, k)(n,k) with 1 ℓ ≥ 1 ℓ >= 1\ell \geq 1ℓ≥1, then there exists F F F ′ ⊆ F F^(')sube FF^{\prime} \subseteq FF′⊆F which is ℓ ℓ\ellℓ-dense ( n , k ) ( n , k ) (n,k)(n, k)(n,k) such that F F FFF is p p ppp-homogeneous.
Indeed, (ii)' is trivial from the definition of density. For (i)', define c : [ F ] 2 2 c : [ F ] 2 → 2 c:[F]^(2)rarr2c:[F]^{2} \rightarrow 2c:[F]2→2 as c ( { x , y } ) = c ( { x , y } ) = c({x,y})=c(\{x, y\})=c({x,y})= 0 [ x , y ) N E = 0 ↔ [ x , y ) N ∩ E = ∅ 0harr[x,y)_(N)nn E=O/0 \leftrightarrow[x, y)_{\mathbb{N}} \cap E=\emptyset0↔[x,y)N∩E=∅, and take a c c ccc-homogeneous set F F F ′ ⊆ F F^(')sube FF^{\prime} \subseteq FF′⊆F such that F F ′ F^(')F^{\prime}F′ is ℓ ℓ\ellℓ-dense ( n , k ) ( n , k ) (n,k)(n, k)(n,k) If [ F ] 2 c 1 ( 1 ) F ′ 2 ⊆ c − 1 ( 1 ) [F^(')]^(2)subec^(-1)(1)\left[F^{\prime}\right]^{2} \subseteq c^{-1}(1)[F′]2⊆c−1(1), then put F = F { min F } F ′ ′ = F ′ ∖ min F ′ F^('')=F^(')\\{minF^(')}F^{\prime \prime}=F^{\prime} \backslash\left\{\min F^{\prime}\right\}F′′=F′∖{minF′} and we have | F | | E | min E < min F F ′ ′ ≤ | E | ≤ min E < min F ′ ′ |F^('')| <= |E| <= min E < minF^('')\left|F^{\prime \prime}\right| \leq|E| \leq \min E<\min F^{\prime \prime}|F′′|≤|E|≤minE<minF′′, but F F ′ ′ F^('')F^{\prime \prime}F′′ must be relatively large since it is at least 0 -dense ( n , k ) ( n , k ) (n,k)(n, k)(n,k). Hence [ F ] 2 c 1 ( 0 ) F ′ 2 ⊆ c − 1 ( 0 ) [F^(')]^(2)subec^(-1)(0)\left[F^{\prime}\right]^{2} \subseteq c^{-1}(0)[F′]2⊆c−1(0), which we are done.
For the next theorem, we want to formalize model-theoretic arguments within second-order arithmetic. Within W K L 0 W K L 0 WKL_(0)\mathrm{WKL}_{0}WKL0, one can set up basic (countable) model theory for first-order logic, and then prove Gödel's completeness theorem [39, SECTIONS II. 8 AND IV.3]. Standard techniques for countable nonstandard models of arithmetic such as the compactness theorem, over/underspill, back and forth, recursive saturation and forcing are naturally formalizable once a countable model with a full evaluation function (truth definition) is provided. On the other hand, it is not possible in general to consider N N N\mathbb{N}N itself as a model of first-order arithmetic since its truth definition is too complicated, 10 10 ^(10){ }^{10}10 hence it is not easy to guarantee that a family of true sentences are consistent. Still, we can deal with the consistency of Π 2 Π 2 Pi_(2)\Pi_{2}Π2-sentences as follows.
Lemma 5.4. R C A 0 R C A 0 RCA_(0)\mathrm{RCA}_{0}RCA0 proves the following. Let A = A 1 , , A k A → = A 1 , … , A k vec(A)=A_(1),dots,A_(k)\vec{A}=A_{1}, \ldots, A_{k}A→=A1,…,Ak be sets, and let Γ Î“ Gamma\GammaΓ be a set of true Π 2 A Π 2 A → Pi_(2)^( vec(A))\Pi_{2}^{\vec{A}}Π2A→-sentences. Then, Γ Î“ Gamma\GammaΓ is consistent (with considering A A → vec(A)\vec{A}A→ as second-order constants). 11 11 ^(11){ }^{11}11 Proof. We work within R C A 0 R C A 0 RCA_(0)\mathrm{RCA}_{0}RCA0 and show that N N N\mathbb{N}N (together with A A → vec(A)\vec{A}A→ ) is a weak model of Γ Î“ Gamma\GammaΓ in the sense of [39, DEFINITION II.8.9]. It is enough to construct a function f : S Γ 2 f : S Γ → 2 f:S^(Gamma)rarr2f: S^{\Gamma} \rightarrow 2f:SΓ→2 which satisfies Tarski's truth definition, where S Γ S Γ S^(Gamma)S^{\Gamma}SΓ is the set of all substitution instances of subformulas of Γ Î“ Gamma\GammaΓ. Let S 0 Γ S 0 Γ S_(0)^(Gamma)S_{0}^{\Gamma}S0Γ be the set of all substitution instances of Σ 0 A Σ 0 A → Sigma_(0)^( vec(A))\Sigma_{0}^{\vec{A}}Σ0A→-subformulas of Γ Î“ Gamma\GammaΓ. Since there is a Π 1 A Π 1 A → − Pi_(1)^( vec(A))-\Pi_{1}^{\vec{A}}-Π1A→− formula which defines the truth of all Σ 0 A Σ 0 A → Sigma_(0)^( vec(A))\Sigma_{0}^{\vec{A}}Σ0A→-formulas, one can take a function f : S 0 Γ 2 f : S 0 Γ → 2 f:S_(0)^(Gamma)rarr2f: S_{0}^{\Gamma} \rightarrow 2f:S0Γ→2 which satisfies the truth definition. Then f f fff can be expanded to S Γ S Γ S^(Gamma)S^{\Gamma}SΓ by putting the truth value 1 for all sentences in S Γ S 0 Γ S Γ ∖ S 0 Γ S^(Gamma)\\S_(0)^(Gamma)S^{\Gamma} \backslash S_{0}^{\Gamma}SΓ∖S0Γ. (They are Σ 1 A Σ 1 A → Sigma_(1)^( vec(A))\Sigma_{1}^{\vec{A}}Σ1A→ or Π 2 A Π 2 A → Pi_(2)^( vec(A))\Pi_{2}^{\vec{A}}Π2A→ and always true.)
Theorem 5.5. Let T R C A 0 T ⊇ R C A 0 T supeRCA_(0)T \supseteq \mathrm{RCA}_{0}T⊇RCA0 be an L 2 L 2 L_(2)\mathscr{L}_{2}L2-theory, and let Y Y YYY be an indicator for T T TTT.
  1. For any r Π 1 1 r Π 1 1 rPi_(1)^(1)\mathrm{r} \Pi_{1}^{1}rΠ11-sentence φ , T φ φ , T ⊢ φ varphi,T|--varphi\varphi, T \vdash \varphiφ,T⊢φ if and only if R C A 0 + { Y m : m N } φ R C A 0 + { Y ≥ m : m ∈ N } ⊢ φ RCA_(0)+{Y >= m:m inN}|--varphi\mathrm{RCA}_{0}+\{Y \geq m: m \in \mathfrak{N}\} \vdash \varphiRCA0+{Y≥m:m∈N}⊢φ.
  2. For any Π 2 Π 2 Pi_(2)\Pi_{2}Π2-sentence φ , T φ φ , T ⊢ φ varphi,T|--varphi\varphi, T \vdash \varphiφ,T⊢φ if and only if Σ 1 + { Y i n t m : m } φ . 12 ∣ Σ 1 + Y i n t ≥ m : m ∈ ℜ ⊢ φ . 12 ∣Sigma_(1)+{Y^(int) >= m:m inℜ}|--varphi.^(12)\mid \Sigma_{1}+\left\{Y^{\mathrm{int}} \geq m: m \in \Re\right\} \vdash \varphi .{ }^{12}∣Σ1+{Yint≥m:m∈ℜ}⊢φ.12
If Y Y YYY is an indicator for T T TTT provably in W K L 0 W K L 0 WKL_(0)\mathrm{WKL}_{0}WKL0, we also have the following:
  1. Over R C A 0 , r Π 1 1 corr ( T ) R C A 0 , r Π 1 1 − corr ⁡ ( T ) RCA_(0),rPi_(1)^(1)-corr(T)\mathrm{RCA}_{0}, \mathrm{r} \Pi_{1}^{1}-\operatorname{corr}(T)RCA0,rΠ11−corr⁡(T) is equivalent to m Y m ∀ m Y ≥ m AA mY >= m\forall m Y \geq m∀mY≥m.
  2. Over Σ 1 , Π 2 ∣ Σ 1 , Π 2 ∣Sigma_(1),Pi_(2)\mid \Sigma_{1}, \Pi_{2}∣Σ1,Π2-corr( T ) T {:T)\left.T\right)T) is equivalent to m Y i n t m ∀ m Y i n t ≥ m AA mY^(int) >= m\forall m Y^{\mathrm{int}} \geq m∀mYint≥m.
Proof. We show statements 1 and 3. (Statements 2 and 4 can be shown similarly.)
The right-to-left direction of statement 1 follows from Theorem 5.1.1 and Tanaka's self-embedding theorem [42]. Indeed, if ( M , S ) ( M , S ) (M,S)(M, S)(M,S) is a countable nonstandard model of T T TTT, then there exists a model ( M ¯ , S ¯ ) ( M ¯ , S ¯ ) ( bar(M), bar(S))(\bar{M}, \bar{S})(M¯,S¯) which is isomorphic to ( M , S ) ( M , S ) (M,S)(M, S)(M,S) such that M e M ¯ M ⊊ e M ¯ M⊊e bar(M)M \subsetneq e \bar{M}M⊊eM¯ and S Cod ( M ¯ / M ) S ⊆ Cod ⁡ ( M ¯ / M ) S sube Cod( bar(M)//M)S \subseteq \operatorname{Cod}(\bar{M} / M)S⊆Cod⁡(M¯/M). If X S X ∈ S X in SX \in SX∈S is infinite in ( M , S ) ( M , S ) (M,S)(M, S)(M,S) and m m ∈ ℜ m inℜm \in \Rem∈ℜ, then there exists F ¯ [ M ¯ ] < M ¯ F ¯ ∈ [ M ¯ ] < M ¯ bar(F)in[ bar(M)]^( < bar(M))\bar{F} \in[\bar{M}]^{<\bar{M}}F¯∈[M¯]<M¯ such that X = F ¯ M X = F ¯ ∩ M X= bar(F)nn MX=\bar{F} \cap MX=F¯∩M. By the condition (cut), Y ( F ¯ ) m Y ( F ¯ ) ≥ m Y( bar(F)) >= mY(\bar{F}) \geq mY(F¯)≥m, hence there exists a set F [ M ] < M F ∈ [ M ] < M F in[M]^( < M)F \in[M]^{<M}F∈[M]<M such that F X F ⊆ X F sube XF \subseteq XF⊆X and Y ( F ) m Y ( F ) ≥ m Y(F) >= mY(F) \geq mY(F)≥m by underspill.
For the left-to-right direction of statement 1 , it is enough to show that if
{ x y θ ( U , x , y ) } R C A 0 { Y m : m } { ∀ x ∃ y θ ( U , x , y ) } ∪ R C A 0 ∪ { Y ≥ m : m ∈ ℜ } {AA x EE y theta(U,x,y)}uuRCA_(0)uu{Y >= m:m inℜ}\{\forall x \exists y \theta(U, x, y)\} \cup \mathrm{RCA}_{0} \cup\{Y \geq m: m \in \Re\}{∀x∃yθ(U,x,y)}∪RCA0∪{Y≥m:m∈ℜ}
is consistent with a second-order constant U U UUU and a Σ 0 U Σ 0 U Sigma_(0)^(U)\Sigma_{0}^{U}Σ0U-formula θ θ theta\thetaθ, then
{ x y θ ( U , x , y ) } T { ∀ x ∃ y θ ( U , x , y ) } ∪ T {AA x EE y theta(U,x,y)}uu T\{\forall x \exists y \theta(U, x, y)\} \cup T{∀x∃yθ(U,x,y)}∪T
is consistent. Let ( M , S ) ( M , S ) (M,S)(M, S)(M,S) be a countable nonstandard model of
{ x y θ ( U , x , y ) } R C A 0 { Y m : m N } { ∀ x ∃ y θ ( U , x , y ) } ∪ R C A 0 ∪ { Y ≥ m : m ∈ N } {AA x EE y theta(U,x,y)}uuRCA_(0)uu{Y >= m:m inN}\{\forall x \exists y \theta(U, x, y)\} \cup \mathrm{RCA}_{0} \cup\{Y \geq m: m \in \mathfrak{N}\}{∀x∃yθ(U,x,y)}∪RCA0∪{Y≥m:m∈N}
Then there exists an infinite set A A AAA in ( M , S ) ( M , S ) (M,S)(M, S)(M,S) such that for any a , b A a , b ∈ A a,b in Aa, b \in Aa,b∈A with a < b a < b a < ba<ba<b, x < a y < b θ ( U , x , y ) ∀ x < a ∃ y < b θ ( U , x , y ) AA x < a EE y < b theta(U,x,y)\forall x<a \exists y<b \theta(U, x, y)∀x<a∃y<bθ(U,x,y). By overspill, there exists an M M MMM-finite set F A F ⊆ A F sube AF \subseteq AF⊆A with Y ( F ) m Y ( F ) ≥ m Y(F) >= mY(F) \geq mY(F)≥m
11 This lemma also follows from (the relativization of) the fact that I Σ 1 I Σ 1 ISigma_(1)I \Sigma_{1}IΣ1 is equivalent to Π 3 Π 3 Pi_(3)\Pi_{3}Π3-corr(EFA). See [1].
12 For statements 1 and 2, the base theories R C A 0 R C A 0 RCA_(0)\mathrm{RCA}_{0}RCA0 and I Σ 1 I Σ 1 ISigma_(1)I \Sigma_{1}IΣ1 can be weakened to R C A 0 R C A 0 ∗ RCA_(0)^(**)\mathrm{RCA}_{0}^{*}RCA0∗ and E F A + B Σ 1 E F A + B Σ 1 EFA+BSigma_(1)\mathrm{EFA}+\mathrm{B} \Sigma_{1}EFA+BΣ1 (the proof still works using recursively saturated models).
for any m m ∈ ℜ m inℜm \in \Rem∈ℜ. By (cut), take I e M I ⊊ e M I⊊eMI \subsetneq e MI⊊eM and S Cod ( M / I ) S ′ ⊆ Cod ⁡ ( M / I ) S^(')sube Cod(M//I)S^{\prime} \subseteq \operatorname{Cod}(M / I)S′⊆Cod⁡(M/I) such that ( I , S ) T I , S ′ ⊨ T (I,S^('))|==T\left(I, S^{\prime}\right) \models T(I,S′)⊨T and F I F ∩ I F nn IF \cap IF∩I is unbounded in I I III. The latter implies ( I , S ) x y θ ( U , x , y ) I , S ′ ⊨ ∀ x ∃ y θ ( U , x , y ) (I,S^('))|==AA x EE y theta(U,x,y)\left(I, S^{\prime}\right) \models \forall x \exists y \theta(U, x, y)(I,S′)⊨∀x∃yθ(U,x,y).
For the left-to-right direction of statement 3 , we first formalize the right-to-left direction of statement 1 within W K L 0 W K L 0 WKL_(0)\mathrm{WKL}_{0}WKL0. In other words, "for each m N , Y m m ∈ N , Y ≥ m m inN,Y >= mm \in \mathbb{N}, Y \geq mm∈N,Y≥m is provable in T T TTT ", is provable within W K L 0 W K L 0 WKL_(0)\mathrm{WKL}_{0}WKL0. Thus it is provable within R C A 0 R C A 0 RCA_(0)\mathrm{RCA}_{0}RCA0 by Theorem 5.1.2 since it is a Π 2 0 Π 2 0 Pi_(2)^(0)\Pi_{2}^{0}Π20-statement, and hence r Π 1 1 r Π 1 1 rPi_(1)^(1)\mathrm{r} \Pi_{1}^{1}rΠ11-corr ( T ) ( T ) (T)(T)(T) implies m Y m ∀ m Y ≥ m AA mY >= m\forall m Y \geq m∀mY≥m.
For the right-to-left direction, again we first work within W K L 0 W K L 0 WKL_(0)\mathrm{WKL}_{0}WKL0. It is enough to show that if x y θ ( U , x , y ) ∀ x ∃ y θ ( U , x , y ) AA x EE y theta(U,x,y)\forall x \exists y \theta(U, x, y)∀x∃yθ(U,x,y) holds for some set U U UUU and a Σ 0 U Σ 0 U Sigma_(0)^(U)\Sigma_{0}^{U}Σ0U-formula θ θ theta\thetaθ, then { x y θ ( U , x , y ) } T { ∀ x ∃ y θ ( U , x , y ) } ∪ T {AA x EE y theta(U,x,y)}uu T\{\forall x \exists y \theta(U, x, y)\} \cup T{∀x∃yθ(U,x,y)}∪T is consistent. Take an infinite set A A AAA such that A A AAA is Δ 1 U Δ 1 U Delta_(1)^(U)\Delta_{1}^{U}Δ1U-definable and for any a , b A a , b ∈ A a,b in Aa, b \in Aa,b∈A with a < b , x < a y < b θ ( U , x , y ) a < b , ∀ x < a ∃ y < b θ ( U , x , y ) a < b,AA x < a EE y < b theta(U,x,y)a<b, \forall x<a \exists y<b \theta(U, x, y)a<b,∀x<a∃y<bθ(U,x,y). Then, by the assumption, for any m N m ∈ N m inNm \in \mathbb{N}m∈N, there exists a finite set F A F ⊆ A F sube AF \subseteq AF⊆A such that Y ( F ) m Y ( F ) ≥ m Y(F) >= mY(F) \geq mY(F)≥m. Thus, by Lemma 5.4, a set of Π 2 U Π 2 U Pi_(2)^(U)\Pi_{2}^{U}Π2U-sentences Γ = E F A U { a F b F ( a < b x < a y < b θ ( U , x , y ) ) } { Y ( F ) m : m N } Γ = E F A U ∪ { ∀ a ∈ F ∀ b ∈ F ( a < b → ∀ x < a ∃ y < b θ ( U , x , y ) ) } ∪ { Y ( F ) ≥ m : m ∈ N } Gamma=EFA^(U)uu{AA a in F AA b in F(a < b rarr AA x < a EE y < b theta(U,x,y))}uu{Y(F) >= m:m inN}\Gamma=\mathrm{EFA}^{U} \cup\{\forall a \in F \forall b \in F(a<b \rightarrow \forall x<a \exists y<b \theta(U, x, y))\} \cup\{Y(F) \geq m: m \in \mathbb{N}\}Γ=EFAU∪{∀a∈F∀b∈F(a<b→∀x<a∃y<bθ(U,x,y))}∪{Y(F)≥m:m∈N} is consistent (consider F F FFF as a new number constant). Take a countable nonstandard model of Γ Î“ Gamma\GammaΓ and formalize the argument for the left-to-right direction of statement 1 , then we see that { x y θ ( U , x , y ) } T { ∀ x ∃ y θ ( U , x , y ) } ∪ T {AA x EE y theta(U,x,y)}uu T\{\forall x \exists y \theta(U, x, y)\} \cup T{∀x∃yθ(U,x,y)}∪T is consistent.
The above argument actually showed that for any set U U UUU, " m Y m ∀ m Y ≥ m AA mY >= m\forall m Y \geq m∀mY≥m with respect to any infinite set A T U A ≤ T U ′ ′ A <= _(T)U^('')A \leq_{T} U^{\prime \prime}A≤TU′′ implies Σ 2 U Σ 2 U Sigma_(2)^(U)\Sigma_{2}^{U}Σ2U-corr ( T ) ( T ) (T)(T)(T). This is a Π 1 1 Π 1 1 Pi_(1)^(1)\Pi_{1}^{1}Π11-statement provable in W K L 0 W K L 0 WKL_(0)W K L_{0}WKL0, so it is also provable within R C A 0 R C A 0 RCA_(0)\mathrm{RCA}_{0}RCA0 by Theorem 5.1.2. Thus R C A 0 R C A 0 RCA_(0)\mathrm{RCA}_{0}RCA0 proves that m Y m ∀ m Y ≥ m AA mY >= m\forall m Y \geq m∀mY≥m implies r Π 1 1 corr ( T ) r Π 1 1 − corr ⁡ ( T ) rPi_(1)^(1)-corr(T)\mathrm{r} \Pi_{1}^{1}-\operatorname{corr}(T)rΠ11−corr⁡(T).
Theorems 5.3 and 5.5 directly connect P H P H PH\mathrm{PH}PH and the correctness statements, and Theorems 2.1 and 2.2 are direct consequences of them. They also imply conservation theorems. Indeed, Theorem 2.6.2 is a direct consequence of Theorems 5.3 and 5.5 plus Theorem 3.8.5 (see [ 25 ] [ 25 ] [25][25][25] ).
Proofs of Theorems 3.2-3.6. By definitions, P H ¯ n , P H ¯ P H ¯ n , P H ¯ bar(PH)^(n), bar(PH)\overline{\mathrm{PH}}^{n}, \overline{\mathrm{PH}}PH¯n,PH¯, and I t P H ¯ k n I t P H ¯ k n It bar(PH)_(k)^(n)\mathrm{It} \overline{\mathrm{PH}}_{k}^{n}ItPH¯kn are equivalent to

variants of P H P H PH\mathrm{PH}PH and corresponding r Π 1 1 r Π 1 1 rPi_(1)^(1)\mathrm{r} \Pi_{1}^{1}rΠ11-correctness statements ( 1 3 ( 1 ↔ 3 (1harr3(1 \leftrightarrow 3(1↔3 and 2 4 2 ↔ 4 2harr42 \leftrightarrow 42↔4 of Theorems 3.2 and 3.3 , 1 2 3.3 , 1 ↔ 2 3.3,1harr23.3,1 \leftrightarrow 23.3,1↔2 of Theorem 3.4 , 1 2 3 3.4 , 1 ↔ 2 ↔ 3 3.4,1harr2harr33.4,1 \leftrightarrow 2 \leftrightarrow 33.4,1↔2↔3 of Theorem 3.5 and Theorem 3.6) follow from Theorems 5.3 and 5.5. Implications between variants of P H P H PH\mathrm{PH}PH and well-foundedness statements follow from Theorem 3.8 (see the paragraph below Theorem 3.8). Other implications can be shown as follows: 3 5 3 → 5 3rarr53 \rightarrow 53→5 of Theorem 3.3 and 2 3 2 → 3 2rarr32 \rightarrow 32→3 of Theorem 3.4 are implied from the formalization of the fact that R C A 0 + I Σ n 0 R C A 0 + I Σ n 0 RCA_(0)+ISigma_(n)^(0)\mathrm{RCA}_{0}+\mathrm{I} \Sigma_{n}^{0}RCA0+IΣn0 proves well-foundedness of ω n k ω n k omega_(n)^(k)\omega_{n}^{k}ωnk for each k k ∈ ℜ k inℜk \in \Rek∈ℜ, and 3 4 3 ↔ 4 3harr43 \leftrightarrow 43↔4 of Theorem 3.3 is implied from the formalization of the conservation result for W K L 0 + R T 2 W K L 0 + R T 2 WKL_(0)+RT^(2)\mathrm{WKL}_{0}+\mathrm{RT}^{2}WKL0+RT2 in [40].

5.3. Indicators corresponding to largeness notions

To obtain a characterization of r Π 2 1 r Π 2 1 rPi_(2)^(1)\mathrm{r} \Pi_{2}^{1}rΠ21-correctness, we modify Theorem 5.5 using indicators which can preserve largeness notions.
Given two finite sets F 0 = { x 0 < < x 1 } F 0 = x 0 < ⋯ < x â„“ − 1 F_(0)={x_(0) < cdots < x_(â„“-1)}F_{0}=\left\{x_{0}<\cdots<x_{\ell-1}\right\}F0={x0<⋯<xℓ−1} and F 1 = { x 0 < < x 1 } F 1 = x 0 ′ < ⋯ < x â„“ ′ − 1 ′ F_(1)={x_(0)^(') < cdots < x_(â„“^(')-1)^(')}F_{1}=\left\{x_{0}^{\prime}<\cdots<x_{\ell^{\prime}-1}^{\prime}\right\}F1={x0′<⋯<xℓ′−1′}, define F 0 F 1 F 0 ⊴ F 1 F_(0)⊴F_(1)F_{0} \unlhd F_{1}F0⊴F1 as â„“ ≤ â„“ ′ â„“ <= â„“^(')\ell \leq \ell^{\prime}ℓ≤ℓ′ and x i x i x i ≥ x i ′ x_(i) >= x_(i)^(')x_{i} \geq x_{i}^{\prime}xi≥xi′ for any i < i < â„“ i < â„“i<\elli<â„“. A prelargeness notion L L L\mathbb{L}L is said to be normal if F 0 L F 0 ∈ L F_(0)inLF_{0} \in \mathbb{L}F0∈L and F 0 F 1 F 0 ⊴ F 1 F_(0)⊴F_(1)F_{0} \unlhd F_{1}F0⊴F1 implies F 1 L F 1 ∈ L F_(1)inLF_{1} \in \mathbb{L}F1∈L. It is not difficult to check that L ω L ω L_(omega)\mathbb{L}_{\omega}Lω is a normal largeness
notion. For a given prelargeness notion L L L\mathbb{L}L, put L + = { F L : G [ 0 , max F ] N ( G F L + = F ∈ L : ∀ G ⊆ [ 0 , max F ] N ( G ⊵ F → L^(+)={F inL:AA G sube[0,max F]_(N)(G⊵F rarr:}\mathbb{L}^{+}=\left\{F \in \mathbb{L}: \forall G \subseteq[0, \max F]_{\mathbb{N}}(G \unrhd F \rightarrow\right.L+={F∈L:∀G⊆[0,maxF]N(G⊵F→ G L ) } G ∈ L ) } G inL)}G \in \mathbb{L})\}G∈L)}. Then L + L + L^(+)\mathbb{L}^{+}L+is a normal prelargeness notion.
Lemma 5.6. The following is provable within W K L 0 W K L 0 WKL_(0)\mathrm{WKL}_{0}WKL0. For any largeness notion L , L + L , L + L,L^(+)\mathbb{L}, \mathbb{L}^{+}L,L+is a largeness notion.
Proof. Assume that L L L\mathbb{L}L is a prelargeness notion and there exists an infinite set X = { x 0 < X = x 0 < X={x_(0) < :}X=\left\{x_{0}<\right.X={x0< x 1 < } x 1 < ⋯ {:x_(1) < cdots}\left.x_{1}<\cdots\right\}x1<⋯} such that no finite subset of X X XXX is a member of L + L + L^(+)\mathbb{L}^{+}L+. Define a tree T N < N T ⊆ N < N T subeN < NT \subseteq \mathbb{N}<\mathbb{N}T⊆N<N as σ T σ ∈ T sigma in T\sigma \in Tσ∈T if and only if σ σ sigma\sigmaσ is strictly increasing, { σ ( i ) : i < | σ | } { x i : i < | σ | } { σ ( i ) : i < | σ | } ⊵ x i : i < | σ | {sigma(i):i < |sigma|}⊵{x_(i):i < |sigma|}\{\sigma(i): i<|\sigma|\} \unrhd\left\{x_{i}: i<|\sigma|\right\}{σ(i):i<|σ|}⊵{xi:i<|σ|} and { σ ( i ) : i < | σ | } L { σ ( i ) : i < | σ | } ∉ L {sigma(i):i < |sigma|}!inL\{\sigma(i): i<|\sigma|\} \notin \mathbb{L}{σ(i):i<|σ|}∉L. Then, T T TTT is a bounded tree and T T TTT is infinite. Take a path h [ T ] h ∈ [ T ] h in[T]h \in[T]h∈[T], then Y = { h ( i ) : i N } Y = { h ( i ) : i ∈ N } Y={h(i):i inN}Y=\{h(i): i \in \mathbb{N}\}Y={h(i):i∈N} is an infinite set and any finite subset of Y Y YYY is not a member of L L L\mathbb{L}L.
Now we generalize the notion of semiregularity with a normal (pre)largeness notion and consider a variant of Theorem 5.5.
Definition 5.2 ( L L L\mathbb{L}L-semiregularity). Let M M MMM be a nonstandard model of E F A L E F A L EFA^(L)E F A^{\mathbb{L}}EFAL, and let L L L\mathbb{L}L be a normal prelargeness notion in M M MMM. Then, a cut I e M I ⊆ e M Isube_(e)MI \subseteq_{e} MI⊆eM is said to be L L L\mathbb{L}L-semiregular if for any finite set F L , F I F ∉ L , F ∩ I F!inL,F nn IF \notin \mathbb{L}, F \cap IF∉L,F∩I is bounded in I I III, or equivalently, L I L ∩ I Lnn I\mathbb{L} \cap IL∩I is a normal largeness notion in ( I , Cod ( M / I ) ) ( I , Cod ⁡ ( M / I ) ) (I,Cod(M//I))(I, \operatorname{Cod}(M / I))(I,Cod⁡(M/I)).
A Σ 0 U Σ 0 U → Sigma_(0)^( vec(U))\Sigma_{0}^{\vec{U}}Σ0U→-formula Y L Y ( L , F , m ) Y L ≡ Y ( L , F , m ) Y^(L)-=Y(L,F,m)Y^{\mathbb{L}} \equiv Y(\mathbb{L}, F, m)YL≡Y(L,F,m) (where L U L ∈ U → Lin vec(U)\mathbb{L} \in \vec{U}L∈U→ ) is said to be an L L L\mathbb{L}L-semiregular indicator for an L 2 L 2 L_(2)\mathscr{L}_{2}L2-theory T T TTT if for any countable nonstandard model M E F A V M ⊨ E F A V M|==EFA^(V)M \models \mathrm{EFA}^{V}M⊨EFAV with U V U → ⊆ V → vec(U)sube vec(V)\vec{U} \subseteq \vec{V}U→⊆V→ such that L L L\mathbb{L}L is a normal prelargeness notion in M , Y L M , Y L M,Y^(L)M, Y^{\mathbb{L}}M,YL defines an indicator for T T TTT on M M MMM but the condition (cut) replaced by
( L L L\mathbb{L}L-cut) Y ( F ) > m Y ( F ) > m Y(F) > mY(F)>mY(F)>m for any m Ω m ∈ Ω m in Omegam \in \Omegam∈Ω if and only if there exists an L L L\mathbb{L}L-semiregular cut I e M I ⊊ e M I⊊eMI \subsetneq e MI⊊eM and S Cod ( M / I ) S ⊆ Cod ⁡ ( M / I ) S sube Cod(M//I)S \subseteq \operatorname{Cod}(M / I)S⊆Cod⁡(M/I) such that ( I , S ) T , U i M I S ( I , S ) ⊨ T , U i M ∩ I ∈ S (I,S)|==T,U_(i)^(M)nn I in S(I, S) \models T, U_{i}^{M} \cap I \in S(I,S)⊨T,UiM∩I∈S for each U i U U i ∈ U → U_(i)in vec(U)U_{i} \in \vec{U}Ui∈U→ and F I F ∩ I F nn IF \cap IF∩I is unbounded in I I III.
Theorem 5.7. Let T W K L 0 T ⊇ W K L 0 T supeWKL_(0)T \supseteq \mathrm{WKL}_{0}T⊇WKL0 be an L 2 L 2 L_(2)\mathscr{L}_{2}L2-theory, and let Y L Y L Y^(L)Y^{\mathbb{L}}YL be an L L L\mathbb{L}L-semiregular indicator for T T TTT provably in W K L 0 W K L 0 WKL_(0)\mathrm{WKL}_{0}WKL0. Then the following assertions are equivalent over W K L 0 W K L 0 WKL_(0)\mathrm{WKL}_{0}WKL0 :
  1. r Π 2 1 corr ( T ) r Π 2 1 − corr ⁡ ( T ) rPi_(2)^(1)-corr(T)\mathrm{r} \Pi_{2}^{1}-\operatorname{corr}(T)rΠ21−corr⁡(T).
  2. For any L L L\mathbb{L}L, if L L L\mathbb{L}L is a normal largeness notion, then m Y L m ∀ m Y L ≥ m AA mY^(L) >= m\forall m Y^{\mathbb{L}} \geq m∀mYL≥m.
Proof. Implication 1 2 1 → 2 1rarr21 \rightarrow 21→2 follows from the same discussion as the proof for Theorem 5.5. To show 2 1 2 → 1 2rarr12 \rightarrow 12→1, we reason within W K L 0 W K L 0 WKL_(0)\mathrm{WKL}_{0}WKL0 and show that, assuming statement 2 is true, if θ ( U ) θ ( U ) theta(U)\theta(U)θ(U) holds for some set U U UUU and an r Π 1 1 r Π 1 1 rPi_(1)^(1)\mathrm{r} \Pi_{1}^{1}rΠ11-formula θ ( U ) θ ( U ) theta(U)\theta(U)θ(U), then { θ ( U ) } T { θ ( U ) } ∪ T {theta(U)}uu T\{\theta(U)\} \cup T{θ(U)}∪T is consistent. By [34, PROPOSITION 2.5], take a Σ 0 0 Σ 0 0 Sigma_(0)^(0)\Sigma_{0}^{0}Σ00-formula η ( G , F ) η ( G , F ) eta(G,F)\eta(G, F)η(G,F) such that W K L W K L W_(KL)W_{K L}WKL proves
V ( θ ( V ) Z ( Z is infinite F fin Z η ( V [ 0 , max F ] N , F ) ) ) ∀ V θ ( V ) ↔ ∀ Z Z  is infinite  → ∃ F ⊆ fin  Z η V ∩ [ 0 , max F ] N , F AA V(theta(V)harr AA Z(Z" is infinite "rarr EE Fsube_("fin ")Z eta(V nn[0,max F]_(N),F)))\forall V\left(\theta(V) \leftrightarrow \forall Z\left(Z \text { is infinite } \rightarrow \exists F \subseteq_{\text {fin }} Z \eta\left(V \cap[0, \max F]_{\mathbb{N}}, F\right)\right)\right)∀V(θ(V)↔∀Z(Z is infinite →∃F⊆fin Zη(V∩[0,maxF]N,F)))
Define L 0 [ N ] < N L 0 ⊆ [ N ] < N L_(0)sube[N]^( < N)\mathbb{L}_{0} \subseteq[\mathbb{N}]^{<\mathbb{N}}L0⊆[N]<N as G L 0 F G η ( U [ 0 , max F ] N , F ) G ∈ L 0 ↔ ∃ F ⊆ G η U ∩ [ 0 , max F ] N , F G inL_(0)harr EE F sube G eta(U nn[0,max F]_(N),F)G \in \mathbb{L}_{0} \leftrightarrow \exists F \subseteq G \eta\left(U \cap[0, \max F]_{\mathbb{N}}, F\right)G∈L0↔∃F⊆Gη(U∩[0,maxF]N,F), and let L = L 0 + L = L 0 + L=L_(0)^(+)\mathbb{L}=\mathbb{L}_{0}^{+}L=L0+. Since θ ( U ) θ ( U ) theta(U)\theta(U)θ(U) holds, L L L\mathbb{L}L is a normal largeness notion. By assumption, we have Y L m Y L ≥ m Y^(L) >= mY^{\mathbb{L}} \geq mYL≥m for any m N m ∈ N m inNm \in \mathbb{N}m∈N. Thus, by Lemma 5.4 , a set of Π 2 U , L Π 2 U , L Pi_(2)^(U,L)\Pi_{2}^{U, \mathbb{L}}Π2U,L-sentences Γ = E F A U , L { L Γ = E F A U , L ∪ { L Gamma=EFA^(U,L)uu{L\Gamma=\mathrm{EFA}^{U, \mathbb{L}} \cup\{\mathbb{L}Γ=EFAU,L∪{L is a normal prelargeness notion } { G ( G L G η ( U [ 0 , max G ] N , G ) ) } { Y L ( F ) m : m N } } ∪ ∀ G G ∈ L → ∃ G ′ η U ∩ 0 , max G ′ N , G ′ ∪ Y L ( F ) ≥ m : m ∈ N }uu{AA G(G inLrarr EEG^(')eta(U nn[0,maxG^(')]_(N),G^(')))}uu{Y^(L)(F) >= m:m inN}\} \cup\left\{\forall G\left(G \in \mathbb{L} \rightarrow \exists G^{\prime} \eta\left(U \cap\left[0, \max G^{\prime}\right]_{\mathbb{N}}, G^{\prime}\right)\right)\right\} \cup\left\{Y^{\mathbb{L}}(F) \geq m: m \in \mathbb{N}\right\}}∪{∀G(G∈L→∃G′η(U∩[0,maxG′]N,G′))}∪{YL(F)≥m:m∈N} is consistent (consider F F FFF as a new number constant).
Take a countable nonstandard model M Γ M ⊨ Γ M|==GammaM \models \GammaM⊨Γ. Then, M Y L ( F M ) m M ⊨ Y L F M ≥ m M|==Y^(L)(F^(M)) >= mM \models Y^{\mathbb{L}}\left(F^{M}\right) \geq mM⊨YL(FM)≥m for any m N m ∈ N m inNm \in \mathbb{N}m∈N and thus there exists an L L L\mathbb{L}L-semiregular cut I e M I ⊊ e M I⊊eMI \subsetneq e MI⊊eM and S Cod ( M / I ) S ⊆ Cod ⁡ ( M / I ) S sube Cod(M//I)S \subseteq \operatorname{Cod}(M / I)S⊆Cod⁡(M/I) such that U I = U M I S , L I = L M I S U I = U M ∩ I ∈ S , L I = L M ∩ I ∈ S U^(I)=U^(M)nn I in S,L^(I)=L^(M)nn I in SU^{I}=U^{M} \cap I \in S, \mathbb{L}^{I}=\mathbb{L}^{M} \cap I \in SUI=UM∩I∈S,LI=LM∩I∈S and ( I , S ) T ( I , S ) ⊨ T (I,S)|==T(I, S) \models T(I,S)⊨T. Since I I III is L M L M L^(M)\mathbb{L}^{M}LM-semiregular, L I L I L^(I)\mathbb{L}^{I}LI is a largeness notion in ( I , S ) ( I , S ) (I,S)(I, S)(I,S). Since M G ( G L G η ( U [ 0 , max G ] N , G ) ) M ⊨ ∀ G G ∈ L → ∃ G ′ η U ∩ 0 , max G ′ N , G ′ M|==AA G(G inLrarr EEG^(')eta(U nn[0,maxG^(')]_(N),G^(')))M \models \forall G\left(G \in \mathbb{L} \rightarrow \exists G^{\prime} \eta\left(U \cap\left[0, \max G^{\prime}\right]_{\mathbb{N}}, G^{\prime}\right)\right)M⊨∀G(G∈L→∃G′η(U∩[0,maxG′]N,G′)), we have ( I , S ) θ ( U I ) ( I , S ) ⊨ θ U I (I,S)|==theta(U^(I))(I, S) \models \theta\left(U^{I}\right)(I,S)⊨θ(UI).
Theorem 5.8. Let n = 2 , 3 , 4 n = 2 , 3 , 4 … n=2,3,4dotsn=2,3,4 \ldotsn=2,3,4… or n = n = ∞ n=oon=\inftyn=∞ and k = 2 , 3 , 4 , k = 2 , 3 , 4 , … k=2,3,4,dotsk=2,3,4, \ldotsk=2,3,4,… or k = k = ∞ k=ook=\inftyk=∞. Define Σ 0 L Σ 0 L Sigma_(0)^(L)\Sigma_{0}^{\mathbb{L}}Σ0L formula Y I t G P H k n L Y I t G P H k n L Y_(ItGPH_(k)^(n))^(L)Y_{\mathrm{ItGPH}_{k}^{n}}^{\mathbb{L}}YItGPHknL as follows:

within W K L 0 W K L 0 WKL_(0)\mathrm{WKL}_{0}WKL0.
Proof. Essentially the same as the proof for Theorem 5.3.3. We additionally need to show the following (which is an analogous of (i)'):
If L L L\mathbb{L}L is a normal prelargeness notion, F F FFF is + 1 â„“ + 1 â„“+1\ell+1â„“+1-dense ( n , k , L L ω ) n , k , L ∩ L ω (n,k,LnnL_(omega))\left(n, k, \mathbb{L} \cap \mathbb{L}_{\omega}\right)(n,k,L∩Lω) with 1 â„“ ≥ 1 â„“ >= 1\ell \geq 1ℓ≥1 and G G GGG is a finite set such that G L G ∉ L G!inLG \notin \mathbb{L}G∉L, then there exists F F F ′ ⊆ F F^(')sube FF^{\prime} \subseteq FF′⊆F such that F F ′ F^(')F^{\prime}F′ is â„“ â„“\ellâ„“-dense ( n , k , L L ω ) n , k , L ∩ L ω (n,k,LnnL_(omega))\left(n, k, \mathbb{L} \cap \mathbb{L}_{\omega}\right)(n,k,L∩Lω) and [ min F , max F ) N G = min F ′ , max F ′ N ∩ G = ∅ [minF^('),maxF^('))_(N)nn G=O/\left[\min F^{\prime}, \max F^{\prime}\right)_{\mathbb{N}} \cap G=\emptyset[minF′,maxF′)N∩G=∅.
Given , L , F â„“ , L , F â„“,L,F\ell, \mathbb{L}, Fâ„“,L,F and G G GGG as above, define c : [ F ] 2 2 c : [ F ] 2 → 2 c:[F]^(2)rarr2c:[F]^{2} \rightarrow 2c:[F]2→2 as c ( { x , y } ) = 1 [ x , y ) N G c ( { x , y } ) = 1 ↔ [ x , y ) N ∩ G ≠ ∅ c({x,y})=1harr[x,y)_(N)nn G!=O/c(\{x, y\})=1 \leftrightarrow[x, y)_{\mathbb{N}} \cap G \neq \emptysetc({x,y})=1↔[x,y)N∩G≠∅. Take a c c ccc-homogeneous set F F F ′ ⊆ F F^(')sube FF^{\prime} \subseteq FF′⊆F such that F F ′ F^(')F^{\prime}F′ is â„“ â„“\ellâ„“-dense ( n , k , L L ω ) n , k , L ∩ L ω (n,k,LnnL_(omega))\left(n, k, \mathbb{L} \cap \mathbb{L}_{\omega}\right)(n,k,L∩Lω). If [ F ] 2 c 1 ( 0 ) F ′ 2 ⊆ c − 1 ( 0 ) [F^(')]^(2)subec^(-1)(0)\left[F^{\prime}\right]^{2} \subseteq c^{-1}(0)[F′]2⊆c−1(0), we are done, so assume [ F ] 2 c 1 ( 1 ) F ′ 2 ⊆ c − 1 ( 1 ) [F^(')]^(2)subec^(-1)(1)\left[F^{\prime}\right]^{2} \subseteq c^{-1}(1)[F′]2⊆c−1(1). Put G = G [ min F , max F ) N G ′ = G ∩ min F ′ , max F ′ N G^(')=G nn[minF^('),maxF^('))_(N)G^{\prime}=G \cap\left[\min F^{\prime}, \max F^{\prime}\right)_{\mathbb{N}}G′=G∩[minF′,maxF′)N and F = F { min F } F ′ ′ = F ′ ∖ min F ′ F^('')=F^(')\\{minF^(')}F^{\prime \prime}=F^{\prime} \backslash\left\{\min F^{\prime}\right\}F′′=F′∖{minF′}. Then F F ′ ′ F^('')F^{\prime \prime}F′′ is at least 0 -dense ( n , k , L L ω ) n , k , L ∩ L ω (n,k,LnnL_(omega))\left(n, k, \mathbb{L} \cap \mathbb{L}_{\omega}\right)(n,k,L∩Lω) and thus F L F ′ ′ ∈ L F^('')inLF^{\prime \prime} \in \mathbb{L}F′′∈L. On the other hand, G F G ′ ⊵ F ′ ′ G^(')⊵F^('')G^{\prime} \unrhd F^{\prime \prime}G′⊵F′′ by the definition of c c ccc, and thus G L G ′ ∈ L G^(')inLG^{\prime} \in \mathbb{L}G′∈L. This is a contradiction since G G G ′ ⊆ G G^(')sube GG^{\prime} \subseteq GG′⊆G and G L G ∉ L G!inLG \notin \mathbb{L}G∉L.
Proofs of Theorems 4.6, 4.7, and 4.8. By Lemma 5.6, I t G P H k n I t G P H k n ItGPH_(k)^(n)\mathrm{ItGPH}_{k}^{n}ItGPHkn is equivalent to the statement

ACKNOWLEDGMENTS

It is a pleasure to thank Leszek Kołodziejczyk for careful reading of the manuscript and helpful discussions and comments, and to Kazuyuki Tanaka for helpful suggestions and advice.

FUNDING

This work is partially supported by JSPS KAKENHI grant number 19K03601.

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KEITA YOKOYAMA

Mathematical Institute, Tohoku University, 6-3, Aramaki Aza-Aoba, Aoba-ku, Sendai 9808578, Japan, keita.yokoyama.c2@ tohoku.ac.jp

CONSTRAINT SATISFACTION PROBLEM: WHAT MAKES THE PROBLEM EASY

DMITRIY ZHUK

Abstract

The Constraint Satisfaction Problem is the problem of deciding whether there is an assignment to a set of variables subject to some specified constraints. Systems of linear equations, graph coloring, and many other combinatorial problems can be expressed as Constraint Satisfaction Problems for some constraint language. In 1993 it was conjectured that for any constraint language the problem is either solvable in polynomial time, or NPcomplete, and for many years this conjecture was the main open question in the area. After this conjecture was resolved in 2017, we finally can say what makes the problem hard and what makes the problem easy. In the first part of the paper, we give an elementary introduction to the area, explaining how the full classification appeared and why it is formulated in terms of polymorphisms. We discuss what makes the problem NP-hard, what makes the problem solvable by local consistency checking, and explain briefly the main idea of one of the two proofs of the conjecture. The second part of the paper is devoted to the extension of the CSP, called Quantified CSP, where we allow using both universal and existential quantifiers. Finally, we discuss briefly other variants of the CSP, as well as some open questions related to them.

MATHEMATICS SUBJECT CLASSIFICATION 2020

Primary 3B70; Secondary 08A70, 68Q17, 03D15, 68T27, 03B10

KEYWORDS

Constraint satisfaction problem, computational complexity, CSP dichotomy, quantified constraints

1. INTRODUCTION

Probably the main question in theoretical computer science is to understand why some computational problems are easy (solvable in polynomial time) while others are difficult (NP-hard, PSpace-hard, and so on). What is the difference between P and NP? Why a system of linear equations can be solved in polynomial time by the Gaussian elimination but we cannot check whether a graph is 3-colorable in polynomial time (if we believe that P N P ) P ≠ N P ) P!=NP)\mathrm{P} \neq \mathrm{NP})P≠NP). What is the principal difference between these two problems? To work on this question, first we would like to classify the problems by whether they are solvable in polynomial time (tractable) or NP-complete. Even for very simple decision problems, sometimes we do not know the answer.
For example, a system of linear equations in Z 2 Z 2 Z_(2)\mathbb{Z}_{2}Z2 can be solved by Gaussian elimination, but if we are allowed to add one linear equation with usual sum for integers then the problem becomes NP-complete [26]. Surprisingly, the complexity is not known if we can add one equation modulo 24 to a system of linear equations in Z 2 Z 2 Z_(2)\mathbb{Z}_{2}Z2 (variables are still from { 0 , 1 } ) { 0 , 1 } ) {0,1})\{0,1\}){0,1}) [17]. In the paper we give a formal definition to such problems and discuss why some of them can be solved in polynomial time, while others are NP-hard.

2. CONSTRAINT SATISFACTION PROBLEM

The above problems are known as the Constraint Satisfaction Problem (CSP), which is the problem of deciding whether there is an assignment to a set of variables subject to some specified constraints. Formally, the Constraint Satisfaction Problem is defined as a triple X , D , C ⟨ X , D , C ⟩ (:X,D,C:)\langle\mathbf{X}, \mathbf{D}, \mathbf{C}\rangle⟨X,D,C⟩, where
  • X = { x 1 , , x n } X = x 1 , … , x n X={x_(1),dots,x_(n)}\mathbf{X}=\left\{x_{1}, \ldots, x_{n}\right\}X={x1,…,xn} is a set of variables,
  • D = { D 1 , , D n } D = D 1 , … , D n D={D_(1),dots,D_(n)}\mathbf{D}=\left\{D_{1}, \ldots, D_{n}\right\}D={D1,…,Dn} is a set of the respective domains,
  • C = { C 1 , , C m } C = C 1 , … , C m C={C_(1),dots,C_(m)}\mathbf{C}=\left\{C_{1}, \ldots, C_{m}\right\}C={C1,…,Cm} is a set of constraints,
where each variable x i x i x_(i)x_{i}xi can take on values in the nonempty domain D i D i D_(i)D_{i}Di, every constraint C j C C j ∈ C C_(j)inCC_{j} \in \mathbf{C}Cj∈C is a pair ( t j , R j ) t j , R j (t_(j),R_(j))\left(t_{j}, R_{j}\right)(tj,Rj) where t j t j t_(j)t_{j}tj is a tuple of variables of length m j m j m_(j)m_{j}mj, called the constraint scope, and R j R j R_(j)R_{j}Rj is an m j m j m_(j)m_{j}mj-ary relation on the corresponding domains, called the constraint relation.
The question is whether there exists a solution to X , D , C ⟨ X , D , C ⟩ (:X,D,C:)\langle\mathbf{X}, \mathbf{D}, \mathbf{C}\rangle⟨X,D,C⟩, that is, a mapping that assigns a value from D i D i D_(i)D_{i}Di to every variable x i x i x_(i)x_{i}xi such that for each constraint C j C j C_(j)C_{j}Cj the image of the constraint scope is a member of the constraint relation.
To simplify the presentation, we assume that the domain of every variable is a finite set A A AAA. We also assume that all the relations are from a set Γ Î“ Gamma\GammaΓ, which we call the constraint language. Then the Constraint Satisfaction Problem over a constraint language Γ Î“ Gamma\GammaΓ, denoted CSP ( Γ ) CSP ⁡ ( Γ ) CSP(Gamma)\operatorname{CSP}(\Gamma)CSP⁡(Γ), is the following decision problem: given a conjunctive formula
R 1 ( v 1 , 1 , , v 1 , n 1 ) R s ( v s , 1 , , v s , n s ) R 1 v 1 , 1 , … , v 1 , n 1 ∧ ⋯ ∧ R s v s , 1 , … , v s , n s R_(1)(v_(1,1),dots,v_(1,n_(1)))^^cdots^^R_(s)(v_(s,1),dots,v_(s,n_(s)))R_{1}\left(v_{1,1}, \ldots, v_{1, n_{1}}\right) \wedge \cdots \wedge R_{s}\left(v_{s, 1}, \ldots, v_{s, n_{s}}\right)R1(v1,1,…,v1,n1)∧⋯∧Rs(vs,1,…,vs,ns)
where R 1 , , R s Γ R 1 , … , R s ∈ Γ R_(1),dots,R_(s)in GammaR_{1}, \ldots, R_{s} \in \GammaR1,…,Rs∈Γ, and v i , j { x 1 , , x n } v i , j ∈ x 1 , … , x n v_(i,j)in{x_(1),dots,x_(n)}v_{i, j} \in\left\{x_{1}, \ldots, x_{n}\right\}vi,j∈{x1,…,xn} for every i , j i , j i,ji, ji,j, decide whether this formula is satisfiable. Note that in the paper we do not distinguish between relations and predicates, and in the previous formula we write relations meaning predicates.

2.1. Examples

It is well known that many combinatorial problems can be expressed as CSP ( Γ ) CSP ⁡ ( Γ ) CSP(Gamma)\operatorname{CSP}(\Gamma)CSP⁡(Γ) for some constraint language Γ Î“ Gamma\GammaΓ. Moreover, for some Γ Î“ Gamma\GammaΓ the corresponding decision problem can be solved in polynomial time; while for others it is NP-complete. It was conjectured that CSP ( Γ ) CSP ⁡ ( Γ ) CSP(Gamma)\operatorname{CSP}(\Gamma)CSP⁡(Γ) is either in P P P\mathrm{P}P or NP-complete [29]. Let us consider several examples.
System of linear equations. Let A = { 0 , 1 } A = { 0 , 1 } A={0,1}A=\{0,1\}A={0,1} and
Γ = { a 1 x 1 + a 2 x 2 + + a k x k = a 0 a 0 , a 1 , , a k Z 2 } Γ = a 1 x 1 + a 2 x 2 + ⋯ + a k x k = a 0 ∣ a 0 , a 1 , … , a k ∈ Z 2 Gamma={a_(1)x_(1)+a_(2)x_(2)+cdots+a_(k)x_(k)=a_(0)∣a_(0),a_(1),dots,a_(k)inZ_(2)}\Gamma=\left\{a_{1} x_{1}+a_{2} x_{2}+\cdots+a_{k} x_{k}=a_{0} \mid a_{0}, a_{1}, \ldots, a_{k} \in \mathbb{Z}_{2}\right\}Γ={a1x1+a2x2+⋯+akxk=a0∣a0,a1,…,ak∈Z2}
i.e., Γ Î“ Gamma\GammaΓ consists of all linear equations in the field Z 2 Z 2 Z_(2)\mathbb{Z}_{2}Z2. Then CSP ( Γ ) CSP ⁡ ( Γ ) CSP(Gamma)\operatorname{CSP}(\Gamma)CSP⁡(Γ) is equivalent to the problem of solving a system of linear equations, which is solvable by the Gaussian elimination in polynomial time, thus, CSP ( Γ ) CSP ⁡ ( Γ ) CSP(Gamma)\operatorname{CSP}(\Gamma)CSP⁡(Γ) is in P P P\mathrm{P}P.
Graph 2-coloring. To color a graph using two colors, we just need to choose a color of every vertex so that adjacent vertices have different colors. We assign a variable to each vertex, and encode the two colors with 0 and 1. For an edge between the i i iii th and j j jjj th vertices, we add the constraint x i x j x i ≠ x j x_(i)!=x_(j)x_{i} \neq x_{j}xi≠xj. For instance, the 5 -cycle is equivalent to the CSP instance
( x 1 x 2 ) ( x 2 x 3 ) ( x 3 x 4 ) ( x 4 x 5 ) ( x 5 x 1 ) x 1 ≠ x 2 ∧ x 2 ≠ x 3 ∧ x 3 ≠ x 4 ∧ x 4 ≠ x 5 ∧ x 5 ≠ x 1 (x_(1)!=x_(2))^^(x_(2)!=x_(3))^^(x_(3)!=x_(4))^^(x_(4)!=x_(5))^^(x_(5)!=x_(1))\left(x_{1} \neq x_{2}\right) \wedge\left(x_{2} \neq x_{3}\right) \wedge\left(x_{3} \neq x_{4}\right) \wedge\left(x_{4} \neq x_{5}\right) \wedge\left(x_{5} \neq x_{1}\right)(x1≠x2)∧(x2≠x3)∧(x3≠x4)∧(x4≠x5)∧(x5≠x1)
Hence, the problem of graph 2-coloring is equivalent to CSP ( Γ ) CSP ⁡ ( Γ ) CSP(Gamma)\operatorname{CSP}(\Gamma)CSP⁡(Γ) for A = { 0 , 1 } A = { 0 , 1 } A={0,1}A=\{0,1\}A={0,1} and Γ = { } Γ = { ≠ } Gamma={!=}\Gamma=\{\neq\}Γ={≠}. This problem can be solved locally. We choose a color of some vertex, then we color their neighbors with a different color, and so on. Either we will color all the vertices, or we will find an odd cycle, which means that the graph is not colorable using two colors. Thus, this problem is solvable in polynomial time.
Graph 3-coloring. Similarly, the problem of coloring a graph using 3 colors is equivalent to CSP ( Γ ) CSP ⁡ ( Γ ) CSP(Gamma)\operatorname{CSP}(\Gamma)CSP⁡(Γ) for A = { 0 , 1 , 2 } A = { 0 , 1 , 2 } A={0,1,2}A=\{0,1,2\}A={0,1,2} and Γ = { } Γ = { ≠ } Gamma={!=}\Gamma=\{\neq\}Γ={≠}. Unlike the graph 2-coloring, this problem is known to be NP-complete [1].
NAE-satisfability and 1IN3-satisfability. Suppose A = { 0 , 1 } A = { 0 , 1 } A={0,1}A=\{0,1\}A={0,1}. NAE is the ternary notall-equal relation, that is, NAE = { 0 , 1 } 3 { ( 0 , 0 , 0 ) , ( 1 , 1 , 1 ) } = { 0 , 1 } 3 ∖ { ( 0 , 0 , 0 ) , ( 1 , 1 , 1 ) } ={0,1}^(3)\\{(0,0,0),(1,1,1)}=\{0,1\}^{3} \backslash\{(0,0,0),(1,1,1)\}={0,1}3∖{(0,0,0),(1,1,1)}. 1IN3 is the ternary 1-in-3 relation, that is, 1 IN 3 = { ( 0 , 0 , 1 ) , ( 0 , 1 , 0 ) , ( 1 , 0 , 0 ) } 1 IN ⁡ 3 = { ( 0 , 0 , 1 ) , ( 0 , 1 , 0 ) , ( 1 , 0 , 0 ) } 1IN 3={(0,0,1),(0,1,0),(1,0,0)}1 \operatorname{IN} 3=\{(0,0,1),(0,1,0),(1,0,0)\}1IN⁡3={(0,0,1),(0,1,0),(1,0,0)}. As it is known [40], both CSP ( { N A E } ) CSP ⁡ ( { N A E } ) CSP({NAE})\operatorname{CSP}(\{\mathrm{NAE}\})CSP⁡({NAE}) and CSP ( { 1 I N 3 } ) CSP ⁡ ( { 1 I N 3 } ) CSP({1IN3})\operatorname{CSP}(\{1 \mathrm{IN} 3\})CSP⁡({1IN3}) are NP-complete.
The main goal of this paper is to explain why the first two examples are in P P P\mathrm{P}P, while the others are NP-hard.

2.2. Reduction from one language to another

To prove the hardness result, we usually reduce a problem to a known NP-hard problem. Let us show how we can go from one constraint language to another. CSP ( Γ ) CSP ⁡ ( Γ ) CSP(Gamma)\operatorname{CSP}(\Gamma)CSP⁡(Γ) can be
viewed as the problem of evaluating a sentence
(2.1) x 1 x n ( R 1 ( v 1 , 1 , , v 1 , n 1 ) R s ( v s , 1 , , v s , n s ) ) (2.1) ∃ x 1 … ∃ x n R 1 v 1 , 1 , … , v 1 , n 1 ∧ ⋯ ∧ R s v s , 1 , … , v s , n s {:(2.1)EEx_(1)dots EEx_(n)(R_(1)(v_(1,1),dots,v_(1,n_(1)))^^cdots^^R_(s)(v_(s,1),dots,v_(s,n_(s)))):}\begin{equation*} \exists x_{1} \ldots \exists x_{n}\left(R_{1}\left(v_{1,1}, \ldots, v_{1, n_{1}}\right) \wedge \cdots \wedge R_{s}\left(v_{s, 1}, \ldots, v_{s, n_{s}}\right)\right) \tag{2.1} \end{equation*}(2.1)∃x1…∃xn(R1(v1,1,…,v1,n1)∧⋯∧Rs(vs,1,…,vs,ns))
where all variables are existentially quantified. Hence, if we could express one language using conjunctions and existential quantifiers from another language, then we get a reduction from one CSP to another. Let us explain how it works on a concrete example.
Let NA1 = { 0 , 1 } 3 { ( 1 , 1 , 1 ) } = { 0 , 1 } 3 ∖ { ( 1 , 1 , 1 ) } ={0,1}^(3)\\{(1,1,1)}=\{0,1\}^{3} \backslash\{(1,1,1)\}={0,1}3∖{(1,1,1)}, that is, a ternary relation that holds whenever not all elements are 1. Let A = { 0 , 1 } , Γ 1 = { N A 1 , } A = { 0 , 1 } , Γ 1 = { N A 1 , ≠ } A={0,1},Gamma_(1)={NA1,!=}A=\{0,1\}, \Gamma_{1}=\{\mathrm{NA} 1, \neq\}A={0,1},Γ1={NA1,≠}, and Γ 2 = { 1 I N 3 } Γ 2 = { 1 I N 3 } Gamma_(2)={1IN3}\Gamma_{2}=\{1 \mathrm{IN} 3\}Γ2={1IN3}. Let us show that CSP ( Γ 1 ) CSP ⁡ Γ 1 CSP(Gamma_(1))\operatorname{CSP}\left(\Gamma_{1}\right)CSP⁡(Γ1) and CSP ( Γ 2 ) CSP ⁡ Γ 2 CSP(Gamma_(2))\operatorname{CSP}\left(\Gamma_{2}\right)CSP⁡(Γ2) are (polynomially) equivalent. We may check that
(2.2) ( x y ) = u v 1 IN 3 ( x , y , u ) 1 IN 3 ( u , u , v ) (2.2) ( x ≠ y ) = ∃ u ∃ v 1 IN ⁡ 3 ( x , y , u ) ∧ 1 IN ⁡ 3 ( u , u , v ) {:(2.2)(x!=y)=EE u EE v1IN 3(x","y","u)^^1IN 3(u","u","v):}\begin{equation*} (x \neq y)=\exists u \exists v 1 \operatorname{IN} 3(x, y, u) \wedge 1 \operatorname{IN} 3(u, u, v) \tag{2.2} \end{equation*}(2.2)(x≠y)=∃u∃v1IN⁡3(x,y,u)∧1IN⁡3(u,u,v)
If fact, from IIN 3 ( u , u , v ) IIN ⁡ 3 ( u , u , v ) IIN 3(u,u,v)\operatorname{IIN} 3(u, u, v)IIN⁡3(u,u,v) we derive that u = 0 u = 0 u=0u=0u=0, hence x y x ≠ y x!=yx \neq yx≠y. Similarly, we have
NA 1 ( x , y , z ) = x y z x y z 1 IN 3 ( x , y , z ) (2.3) 1 IN 3 ( x , x , x ) 1 IN 3 ( y , y , y ) 1 IN 3 ( z , z , z ) NA ⁡ 1 ( x , y , z ) = ∃ x ′ ∃ y ′ ∃ z ′ ∃ x ′ ′ ∃ y ′ ′ ∃ z ′ ′ 1 IN ⁡ 3 x ′ , y ′ , z ′ (2.3) ∧ 1 IN ⁡ 3 x , x ′ , x ′ ′ ∧ 1 IN ⁡ 3 y , y ′ , y ′ ′ ∧ 1 IN ⁡ 3 z , z ′ , z ′ ′ {:[NA 1(x","y","z)=EEx^(')EEy^(')EEz^(')EEx^('')EEy^('')EEz^('')1IN 3(x^('),y^('),z^('))],[(2.3)^^1IN 3(x,x^('),x^(''))^^1IN 3(y,y^('),y^(''))^^1IN 3(z,z^('),z^(''))]:}\begin{align*} \operatorname{NA} 1(x, y, z)=\exists & x^{\prime} \exists y^{\prime} \exists z^{\prime} \exists x^{\prime \prime} \exists y^{\prime \prime} \exists z^{\prime \prime} 1 \operatorname{IN} 3\left(x^{\prime}, y^{\prime}, z^{\prime}\right) \\ & \wedge 1 \operatorname{IN} 3\left(x, x^{\prime}, x^{\prime \prime}\right) \wedge 1 \operatorname{IN} 3\left(y, y^{\prime}, y^{\prime \prime}\right) \wedge 1 \operatorname{IN} 3\left(z, z^{\prime}, z^{\prime \prime}\right) \tag{2.3} \end{align*}NA⁡1(x,y,z)=∃x′∃y′∃z′∃x′′∃y′′∃z′′1IN⁡3(x′,y′,z′)(2.3)∧1IN⁡3(x,x′,x′′)∧1IN⁡3(y,y′,y′′)∧1IN⁡3(z,z′,z′′)
If x = y = z = 1 x = y = z = 1 x=y=z=1x=y=z=1x=y=z=1, then x = y = z = 0 x ′ = y ′ = z ′ = 0 x^(')=y^(')=z^(')=0x^{\prime}=y^{\prime}=z^{\prime}=0x′=y′=z′=0, which contradicts 1 IN 3 ( x , y , z ) 1 IN ⁡ 3 x ′ , y ′ , z ′ 1IN 3(x^('),y^('),z^('))1 \operatorname{IN} 3\left(x^{\prime}, y^{\prime}, z^{\prime}\right)1IN⁡3(x′,y′,z′). In all other cases, we can find an appropriate assignment.
Any instance of CSP ( Γ 1 ) CSP ⁡ Γ 1 CSP(Gamma_(1))\operatorname{CSP}\left(\Gamma_{1}\right)CSP⁡(Γ1) can be reduced to an instance of CSP ( Γ 2 ) CSP ⁡ Γ 2 CSP(Gamma_(2))\operatorname{CSP}\left(\Gamma_{2}\right)CSP⁡(Γ2) in the following way. We replace each constraint ( x i x j ) x i ≠ x j (x_(i)!=x_(j))\left(x_{i} \neq x_{j}\right)(xi≠xj) by the right-hand side of (2.2) introducing two new variables. Also, we replace each constraint NA1 ( x i , x j , x k ) x i , x j , x k (x_(i),x_(j),x_(k))\left(x_{i}, x_{j}, x_{k}\right)(xi,xj,xk) by the right-hand side of (2.3) introducing six new variables. This reduction is obviously polynomial (and even log-space). Similarly, we have
IIN3 ( x , y , z ) = x y z ( NA 1 ( x , y , y ) NA 1 ( y , z , z ) NA 1 ( z , x , x ) ( x x ) ( y y ) ( z z ) NA1 ( x , y , z ) ) IIN3 ⁡ ( x , y , z ) = ∃ x ′ ∃ y ′ ∃ z ′ ( NA ⁡ 1 ( x , y , y ) ∧ NA ⁡ 1 ( y , z , z ) ∧ NA ⁡ 1 ( z , x , x ) ∧ x ≠ x ′ ∧ y ≠ y ′ ∧ z ≠ z ′ ∧ NA1 ⁡ x ′ , y ′ , z ′ {:[IIN3(x","y","z)=EEx^(')EEy^(')EEz^(')(NA 1(x","y","y)^^NA 1(y","z","z)^^NA 1(z","x","x)],[{:^^(x!=x^('))^^(y!=y^('))^^(z!=z^('))^^NA1(x^('),y^('),z^(')))]:}\begin{array}{r} \operatorname{IIN3}(x, y, z)=\exists x^{\prime} \exists y^{\prime} \exists z^{\prime}(\operatorname{NA} 1(x, y, y) \wedge \operatorname{NA} 1(y, z, z) \wedge \operatorname{NA} 1(z, x, x) \\ \left.\wedge\left(x \neq x^{\prime}\right) \wedge\left(y \neq y^{\prime}\right) \wedge\left(z \neq z^{\prime}\right) \wedge \operatorname{NA1}\left(x^{\prime}, y^{\prime}, z^{\prime}\right)\right) \end{array}IIN3⁡(x,y,z)=∃x′∃y′∃z′(NA⁡1(x,y,y)∧NA⁡1(y,z,z)∧NA⁡1(z,x,x)∧(x≠x′)∧(y≠y′)∧(z≠z′)∧NA1⁡(x′,y′,z′))
which implies a polynomial reduction from CSP ( Γ 2 ) CSP ⁡ Γ 2 CSP(Gamma_(2))\operatorname{CSP}\left(\Gamma_{2}\right)CSP⁡(Γ2) to CSP ( Γ 1 ) CSP ⁡ Γ 1 CSP(Gamma_(1))\operatorname{CSP}\left(\Gamma_{1}\right)CSP⁡(Γ1).
Let us give a formal definition for the above reduction. A formula of the form y 1 y n Φ âˆƒ y 1 … ∃ y n Φ EEy_(1)dots EEy_(n)Phi\exists y_{1} \ldots \exists y_{n} \Phi∃y1…∃ynΦ, where Φ Î¦ Phi\PhiΦ is a conjunction of relations from Γ Î“ Gamma\GammaΓ is called a positive primitive formula (pp-formula) over Γ Î“ Gamma\GammaΓ. If R ( x 1 , , x n ) = y 1 y n Φ R x 1 , … , x n = ∃ y 1 … ∃ y n Φ R(x_(1),dots,x_(n))=EEy_(1)dots EEy_(n)PhiR\left(x_{1}, \ldots, x_{n}\right)=\exists y_{1} \ldots \exists y_{n} \PhiR(x1,…,xn)=∃y1…∃ynΦ, then we say that R R RRR is p p p p ppp ppp defined by this formula, and y 1 y n Φ âˆƒ y 1 … ∃ y n Φ EEy_(1)dots EEy_(n)Phi\exists y_{1} \ldots \exists y_{n} \Phi∃y1…∃ynΦ is called its p p p p ppp ppp-definition.
Theorem 2.1 ([35]). Suppose Γ 1 Γ 1 Gamma_(1)\Gamma_{1}Γ1 and Γ 2 Γ 2 Gamma_(2)\Gamma_{2}Γ2 are finite constraint languages such that each relation from Γ 1 Γ 1 Gamma_(1)\Gamma_{1}Γ1 is pp-definable over Γ 2 Γ 2 Gamma_(2)\Gamma_{2}Γ2. Then CSP ( Γ 1 ) CSP ⁡ Γ 1 CSP(Gamma_(1))\operatorname{CSP}\left(\Gamma_{1}\right)CSP⁡(Γ1) is polynomial time reducible to CSP ( Γ 2 ) CSP ⁡ Γ 2 CSP(Gamma_(2))\operatorname{CSP}\left(\Gamma_{2}\right)CSP⁡(Γ2).

2.3. Polymorphisms as invariants

If we can pp-define a relation R R RRR from a constraint language Γ Î“ Gamma\GammaΓ and CSP ( { R } ) CSP ⁡ ( { R } ) CSP({R})\operatorname{CSP}(\{R\})CSP⁡({R}) is NP-hard, then CSP ( Γ ) CSP ⁡ ( Γ ) CSP(Gamma)\operatorname{CSP}(\Gamma)CSP⁡(Γ) is also NP-hard. How to show that such a relation cannot be ppdefined? To prove that something cannot be done, we usually find some fundamental property (invariant) that is satisfied by anything we can obtain. For the relations, the operations play the role of invariants.
We say that an operation f : A n f : A n → f:A^(n)rarrf: A^{n} \rightarrowf:An→ A preserves a relation R R RRR of arity m m mmm if for any tuples ( a 1 , 1 , , a 1 , m ) , , ( a n , 1 , , a n , m ) R a 1 , 1 , … , a 1 , m , … , a n , 1 , … , a n , m ∈ R (a_(1,1),dots,a_(1,m)),dots,(a_(n,1),dots,a_(n,m))in R\left(a_{1,1}, \ldots, a_{1, m}\right), \ldots,\left(a_{n, 1}, \ldots, a_{n, m}\right) \in R(a1,1,…,a1,m),…,(an,1,…,an,m)∈R the tuple
( f ( a 1 , 1 , , a n , 1 ) , , f ( a 1 , m , , a n , m ) ) f a 1 , 1 , … , a n , 1 , … , f a 1 , m , … , a n , m (f(a_(1,1),dots,a_(n,1)),dots,f(a_(1,m),dots,a_(n,m)))\left(f\left(a_{1,1}, \ldots, a_{n, 1}\right), \ldots, f\left(a_{1, m}, \ldots, a_{n, m}\right)\right)(f(a1,1,…,an,1),…,f(a1,m,…,an,m))
is in R R RRR. In this case we also say that f f fff is a polymorphism of R R RRR, and R R RRR is an invariant of f f fff. We say that an operation preserves a set of relations Γ Î“ Gamma\GammaΓ if it preserves every relation in Γ Î“ Gamma\GammaΓ. In this case we also write f f fff is a polymorphism of Γ Î“ Gamma\GammaΓ or f Pol ( Γ ) f ∈ Pol ⁡ ( Γ ) f in Pol(Gamma)f \in \operatorname{Pol}(\Gamma)f∈Pol⁡(Γ). It can be easily checked that if f f fff preserves Γ Î“ Gamma\GammaΓ, then f f fff preserves any relation R R RRR pp-definable from Γ Î“ Gamma\GammaΓ. Moreover, we can show [ 15 , 31 ] [ 15 , 31 ] [15,31][15,31][15,31] that Pol ( Γ 1 ) Pol ( Γ 2 ) Pol ⁡ Γ 1 ⊆ Pol ⁡ Γ 2 Pol(Gamma_(1))sube Pol(Gamma_(2))\operatorname{Pol}\left(\Gamma_{1}\right) \subseteq \operatorname{Pol}\left(\Gamma_{2}\right)Pol⁡(Γ1)⊆Pol⁡(Γ2) if and only if Γ 2 Γ 2 Gamma_(2)\Gamma_{2}Γ2 is pp-definable over Γ 1 Γ 1 Gamma_(1)\Gamma_{1}Γ1, which means that the complexity of CSP ( Γ ) CSP ⁡ ( Γ ) CSP(Gamma)\operatorname{CSP}(\Gamma)CSP⁡(Γ) depends only on Pol ( Γ ) Pol ⁡ ( Γ ) Pol(Gamma)\operatorname{Pol}(\Gamma)Pol⁡(Γ).
Example 1. Let R R RRR be the linear order relation on { 0 , 1 , 2 } { 0 , 1 , 2 } {0,1,2}\{0,1,2\}{0,1,2}, i.e.,
R = ( 0 0 0 1 1 2 0 1 2 1 2 2 ) R = 0      0      0      1      1      2 0      1      2      1      2      2 R=([0,0,0,1,1,2],[0,1,2,1,2,2])R=\left(\begin{array}{llllll} 0 & 0 & 0 & 1 & 1 & 2 \\ 0 & 1 & 2 & 1 & 2 & 2 \end{array}\right)R=(000112012122)
where columns are tuples from the relation. Then "an n n nnn-ary operation f f fff preserves R R RRR " means that for all
( a 1 b 1 ) , , ( a n b n ) R ( a 1 b 1 ) , … , ( a n b n ) ∈ R ((a_(1))/(b_(1))),dots,((a_(n))/(b_(n)))in R\binom{a_{1}}{b_{1}}, \ldots,\binom{a_{n}}{b_{n}} \in R(a1b1),…,(anbn)∈R
that is, a i b i a i ≤ b i a_(i) <= b_(i)a_{i} \leq b_{i}ai≤bi, we have
f ( a 1 a 2 a n b 1 b 2 b n ) := ( f ( a 1 , , a n ) f ( b 1 , , b n ) ) R f a 1      a 2      …      a n b 1      b 2      …      b n := ( f a 1 , … , a n f b 1 , … , b n ) ∈ R f([a_(1),a_(2),dots,a_(n)],[b_(1),b_(2),dots,b_(n)]):=((f(a_(1),dots,a_(n)))/(f(b_(1),dots,b_(n))))in Rf\left(\begin{array}{llll} a_{1} & a_{2} & \ldots & a_{n} \\ b_{1} & b_{2} & \ldots & b_{n} \end{array}\right):=\binom{f\left(a_{1}, \ldots, a_{n}\right)}{f\left(b_{1}, \ldots, b_{n}\right)} \in Rf(a1a2…anb1b2…bn):=(f(a1,…,an)f(b1,…,bn))∈R
that is, f ( a 1 , , a n ) f ( b 1 , , b n ) f a 1 , … , a n ≤ f b 1 , … , b n f(a_(1),dots,a_(n)) <= f(b_(1),dots,b_(n))f\left(a_{1}, \ldots, a_{n}\right) \leq f\left(b_{1}, \ldots, b_{n}\right)f(a1,…,an)≤f(b1,…,bn). In other words, f f fff is monotonic. For instance, the operations max and min are monotonic. By the above observation, we know that any relation pp-definable from R R RRR is also preserved by min and max.
Example 2. Let A = { 0 , 1 } A = { 0 , 1 } A={0,1}A=\{0,1\}A={0,1}. Let us show that 1IN3 cannot be pp-defined from NA1 and x y x ≤ y x <= yx \leq yx≤y. We can check that the conjunction x y x ∧ y x^^yx \wedge yx∧y (an operation on { 0 , 1 } { 0 , 1 } {0,1}\{0,1\}{0,1} ) preserves both NA1 and x y x ≤ y x <= yx \leq yx≤y. However, x y x ∧ y x^^yx \wedge yx∧y does not preserve 1IN3 as we have
( 1 0 0 ) ( 0 1 0 ) = ( 0 0 0 ) 1 I N 3 1 0 0 ∧ 0 1 0 = 0 0 0 ∉ 1 I N 3 ([1],[0],[0])^^([0],[1],[0])=([0],[0],[0])!in1IN3\left(\begin{array}{l} 1 \\ 0 \\ 0 \end{array}\right) \wedge\left(\begin{array}{l} 0 \\ 1 \\ 0 \end{array}\right)=\left(\begin{array}{l} 0 \\ 0 \\ 0 \end{array}\right) \notin 1 \mathrm{IN} 3(100)∧(010)=(000)∉1IN3
For more information on polymorphisms and how they can be used to study the complexity of the CSP, see [6].

2.4. Local consistency

The first step of almost any algorithm solving a CSP instance is checking local consistency. For instance, if a constraint forces a variable to be equal to 0 , then we could substitute 0 and remove this variable.
Suppose we have a CSP instance
(2.4) R 1 ( v 1 , 1 , , v 1 , n 1 ) R s ( v s , 1 , , v s , n s ) (2.4) R 1 v 1 , 1 , … , v 1 , n 1 ∧ ⋯ ∧ R s v s , 1 , … , v s , n s {:(2.4)R_(1)(v_(1,1),dots,v_(1,n_(1)))^^cdots^^R_(s)(v_(s,1),dots,v_(s,n_(s))):}\begin{equation*} R_{1}\left(v_{1,1}, \ldots, v_{1, n_{1}}\right) \wedge \cdots \wedge R_{s}\left(v_{s, 1}, \ldots, v_{s, n_{s}}\right) \tag{2.4} \end{equation*}(2.4)R1(v1,1,…,v1,n1)∧⋯∧Rs(vs,1,…,vs,ns)
This instance is called 1 -consistent (also known as arc-consistent), if for any variable x x xxx any two constraints R i ( v i , 1 , , v i , n i ) R i v i , 1 , … , v i , n i R_(i)(v_(i,1),dots,v_(i,n_(i)))R_{i}\left(v_{i, 1}, \ldots, v_{i, n_{i}}\right)Ri(vi,1,…,vi,ni) and R j ( v j , 1 , , v j , n j ) R j v j , 1 , … , v j , n j R_(j)(v_(j,1),dots,v_(j,n_(j)))R_{j}\left(v_{j, 1}, \ldots, v_{j, n_{j}}\right)Rj(vj,1,…,vj,nj) having this variable in the scope have the same projection onto this variable. This means that for every variable x x xxx there exists D x A D x ⊆ A D_(x)sube AD_{x} \subseteq ADx⊆A, called the domain of x x xxx, such that the projection of any constraint on x x xxx is D x D x D_(x)D_{x}Dx.
Sometimes we need a stronger consistency (similar to singleton-arc-consistency in [36]). We say that z 1 C 1 z 2 C l 1 z l z 1 − C 1 − z 2 − ⋯ − C l − 1 − z l z_(1)-C_(1)-z_(2)-cdots-C_(l-1)-z_(l)z_{1}-C_{1}-z_{2}-\cdots-C_{l-1}-z_{l}z1−C1−z2−⋯−Cl−1−zl is a path in a CSP instance d d ddd if z i , z i + 1 z i , z i + 1 z_(i),z_(i+1)z_{i}, z_{i+1}zi,zi+1 are in the scope of the constraint C i C i C_(i)C_{i}Ci for every i { 1 , 2 , , l 1 } i ∈ { 1 , 2 , … , l − 1 } i in{1,2,dots,l-1}i \in\{1,2, \ldots, l-1\}i∈{1,2,…,l−1}. We say that a a aaa path z 1 C 1 z 2 C l 1 z l z 1 − C 1 − z 2 − ⋯ − C l − 1 − z l z_(1)-C_(1)-z_(2)-cdots-C_(l-1)-z_(l)z_{1}-C_{1}-z_{2}-\cdots-C_{l-1}-z_{l}z1−C1−z2−⋯−Cl−1−zl connects b b bbb and c c ccc if there exist a 1 , a 2 , , a l A a 1 , a 2 , … , a l ∈ A a_(1),a_(2),dots,a_(l)in Aa_{1}, a_{2}, \ldots, a_{l} \in Aa1,a2,…,al∈A such that a 1 = b , a l = c a 1 = b , a l = c a_(1)=b,a_(l)=ca_{1}=b, a_{l}=ca1=b,al=c, and the projection of each C i C i C_(i)C_{i}Ci onto z i , z i + 1 z i , z i + 1 z_(i),z_(i+1)z_{i}, z_{i+1}zi,zi+1 contains the tuple ( a i , a i + 1 ) a i , a i + 1 (a_(i),a_(i+1))\left(a_{i}, a_{i+1}\right)(ai,ai+1). A CSP instance d d ddd is called cycle-consistent if it is 1-consistent and for every variable z z zzz and a D z a ∈ D z a inD_(z)a \in D_{z}a∈Dz any path starting and ending with z z zzz in ℓ ℓ\ellℓ connects a a aaa and a a aaa.
It is not hard to find a polynomial procedure making the instance 1-consistent or cycle-consistent. For 1-consistency, the idea is to find a variable where the consistency is violated, then reduce the domain D x D x D_(x)D_{x}Dx of this variable and reduce the corresponding relations. We repeat this while some constraints violate consistency. Finally, we either get a 1-consistent instance, or we get a contradiction (derive that D x = D x = ∅ D_(x)=O/D_{x}=\varnothingDx=∅ ). For cycle-consistency, we should go deeper. For every variable x x xxx and every value a D x a ∈ D x a inD_(x)a \in D_{x}a∈Dx, we reduce the domain of x x xxx to { a } { a } {a}\{a\}{a} and check whether the remaining instance can be made 1-consistent. If not, then x x xxx cannot be equal to a a aaa, and a a aaa can be excluded from the domain D x D x D_(x)D_{x}Dx.
Later we will show that in some cases 1-consistency and cycle-consistency are enough to solve a CSP instance, that is, any consistent instance has a solution. See [5, 36] for more information about local consistency conditions.

2.5. CSP over a 2-element domain

The complexity of CSP ( Γ ) CSP ⁡ ( Γ ) CSP(Gamma)\operatorname{CSP}(\Gamma)CSP⁡(Γ) for each constraint language Γ Î“ Gamma\GammaΓ on { 0 , 1 } { 0 , 1 } {0,1}\{0,1\}{0,1} was described in 1978 [40]. This classification can be formulated nicely using polymorphisms.
Theorem 2.2 ([34,40]). Suppose A = { 0 , 1 } , Γ A = { 0 , 1 } , Γ A={0,1},GammaA=\{0,1\}, \GammaA={0,1},Γ is a constraint language on A A AAA. Then CSP ( Γ ) CSP ⁡ ( Γ ) CSP(Gamma)\operatorname{CSP}(\Gamma)CSP⁡(Γ) is solvable in polynomial time if
(1) 0 preserves Γ Î“ Gamma\GammaΓ, or
(2) 1 preserves Γ Î“ Gamma\GammaΓ, or
(3) x y x ∨ y x vv yx \vee yx∨y preserves Γ Î“ Gamma\GammaΓ, or
(4) x y x ∧ y x^^yx \wedge yx∧y preserves Γ Î“ Gamma\GammaΓ, or
(5) x y y z x z x y ∨ y z ∨ x z xy vv yz vv xzx y \vee y z \vee x zxy∨yz∨xz preserves Γ Î“ Gamma\GammaΓ, or
(6) x + y + z x + y + z x+y+zx+y+zx+y+z preserves Γ Î“ Gamma\GammaΓ.
CSP ( Γ ) CSP ⁡ ( Γ ) CSP(Gamma)\operatorname{CSP}(\Gamma)CSP⁡(Γ) is NP-complete otherwise.
Let us consider each case and explain how the polymorphisms make the problem easy. Note that the cases (1) and (2), (3) and (4) are dual to each other, that is why we consider only one in each pair in detail.
0 0 0\mathbf{0}0 preserves Γ Î“ Gamma\boldsymbol{\Gamma}Γ. This case is almost trivial. "The constant 0 preserves a relation R Γ R ∈ Γ R in GammaR \in \GammaR∈Γ " means that R ( 0 , 0 , , 0 ) R ( 0 , 0 , … , 0 ) R(0,0,dots,0)R(0,0, \ldots, 0)R(0,0,…,0) holds. If 0 preserves all relations from Γ Î“ Gamma\GammaΓ, then ( 0 , 0 , , 0 ) ( 0 , 0 , … , 0 ) (0,0,dots,0)(0,0, \ldots, 0)(0,0,…,0) is always a solution of a CSP CSP CSP\operatorname{CSP}CSP instance, which makes the problem CSP ( Γ ) CSP ⁡ ( Γ ) CSP(Gamma)\operatorname{CSP}(\Gamma)CSP⁡(Γ) trivial.
x y x ∨ y x vv y\boldsymbol{x} \vee \boldsymbol{y}x∨y preserves Γ Î“ Gamma\GammaΓ. Let us show how to solve an instance of CSP ( Γ ) CSP ⁡ ( Γ ) CSP(Gamma)\operatorname{CSP}(\Gamma)CSP⁡(Γ) if x y Pol ( Γ ) x ∨ y ∈ Pol ⁡ ( Γ ) x vv y in Pol(Gamma)x \vee y \in \operatorname{Pol}(\Gamma)x∨y∈Pol⁡(Γ). First, we make our instance 1-consistent. Then, unless we get a contradiction, every variable x x xxx has its domain D x D x D_(x)D_{x}Dx which is either { 0 } { 0 } {0}\{0\}{0}, or { 1 } { 1 } {1}\{1\}{1}, or { 0 , 1 } { 0 , 1 } {0,1}\{0,1\}{0,1}. We claim that if we send the variables with domain { 0 } { 0 } {0}\{0\}{0} to 0 , and the variables with the domain { 1 } { 1 } {1}\{1\}{1} and { 0 , 1 } { 0 , 1 } {0,1}\{0,1\}{0,1} to 1 , then we get a solution. In fact, if we apply x y x ∨ y x vv yx \vee yx∨y to all the tuples of some constraint, we obtain a tuple consistent with the solution. Thus, 1-consistency guarantees the existence of a solution in this case.
x y y z x z x y ∨ y z ∨ x z xy vv yz vv xz\boldsymbol{x} \boldsymbol{y} \vee \boldsymbol{y} z \vee \boldsymbol{x} zxy∨yz∨xz preserves Γ Î“ Gamma\boldsymbol{\Gamma}Γ. The operation x y y z x z x y ∨ y z ∨ x z xy vv yz vv xzx y \vee y z \vee x zxy∨yz∨xz returns the most popular value and is known as a majority operation. It is not hard to check [2] that any relation preserved by a majority operation can be represented as a conjunction of binary relations, and we may assume that Γ Î“ Gamma\GammaΓ consists of only binary relations. As it is shown in Section 2.8, for a 2-element domain this gives a polynomial algorithm for CSP ( Γ ) CSP ⁡ ( Γ ) CSP(Gamma)\operatorname{CSP}(\Gamma)CSP⁡(Γ). Additionally, we can show [ 36 , 47 ] [ 36 , 47 ] [36,47][36,47][36,47] that any cycle-consistent instance of CSP ( Γ ) CSP ⁡ ( Γ ) CSP(Gamma)\operatorname{CSP}(\Gamma)CSP⁡(Γ) has a solution. Hence to solve an instance, it is sufficient to make it cycle-consistent, and unless we obtain an empty domain (contradiction) the instance has a solution.
x + y + z x + y + z x+y+z\boldsymbol{x}+\boldsymbol{y}+\boldsymbol{z}x+y+z preserves Γ Î“ Gamma\boldsymbol{\Gamma}Γ. It is known (see Lemma 2.8) that x + y + z x + y + z x+y+zx+y+zx+y+z preserves a relation R R RRR if and only if the relation R R RRR can be represented as a conjunction of linear equations. Thus, CSP ( Γ ) CSP ⁡ ( Γ ) CSP(Gamma)\operatorname{CSP}(\Gamma)CSP⁡(Γ) is equivalent to the problem of solving of a system of linear equations in the field Z 2 Z 2 Z_(2)\mathbb{Z}_{2}Z2, which is tractable.

2.6. CSP solvable by local consistency checking

As we see in the previous section all tractable CSPs on a 2-element domain can be solved by two algorithms. The first algorithm just checks some local consistency (1-consistency, cycle-consistency) and, if a sufficient level of consistency achieved, we know that the instance has a solution. The second algorithm is the Gaussian elimination applied to a system of linear equations. In this section we discuss when the first algorithm is sufficient and why some instances can be solved by a local consistency checking, while others require something else.
To simplify the presentation in this section, we assume that all constant relations x = a x = a x=ax=ax=a are in the constraint language. In this case any polymorphism f f fff of Γ Î“ Gamma\GammaΓ is idempotent, that is, f ( x , x , , x ) = x f ( x , x , … , x ) = x f(x,x,dots,x)=xf(x, x, \ldots, x)=xf(x,x,…,x)=x. This restriction does not affect the generality of the results because we can always consider the core of the constraint language and then add all constant relations
(see [34]). Consider the following system of linear equations in Z p Z p Z_(p)\mathbb{Z}_{p}Zp :
(2.5) { x 1 + x 2 = x 3 + 0 x 3 + 0 = x 4 + x 5 x 4 + 0 = x 1 + x 6 x 5 + x 6 = x 2 + 1 (2.5) x 1 + x 2 = x 3 + 0 x 3 + 0 = x 4 + x 5 x 4 + 0 = x 1 + x 6 x 5 + x 6 = x 2 + 1 {:(2.5){[x_(1)+x_(2)=x_(3)+0],[x_(3)+0=x_(4)+x_(5)],[x_(4)+0=x_(1)+x_(6)],[x_(5)+x_(6)=x_(2)+1]:}:}\left\{\begin{array}{l} x_{1}+x_{2}=x_{3}+0 \tag{2.5}\\ x_{3}+0=x_{4}+x_{5} \\ x_{4}+0=x_{1}+x_{6} \\ x_{5}+x_{6}=x_{2}+1 \end{array}\right.(2.5){x1+x2=x3+0x3+0=x4+x5x4+0=x1+x6x5+x6=x2+1
If we calculate the sum of all equations, we will get 0 = 1 0 = 1 0=10=10=1, which means that the system does not have a solution. Nevertheless, we may check that the system is cycle-consistent, which means that the cycle-consistency does not guarantee the existence of a solution for linear equations. In fact, we can show that there does not exist a local consistency condition that guarantees the existence of a solution of a system of linear equations (see [5]).
As it was shown in [ 5 , 47 ] [ 5 , 47 ] [5,47][5,47][5,47] if CSP ( Γ ) CSP ⁡ ( Γ ) CSP(Gamma)\operatorname{CSP}(\Gamma)CSP⁡(Γ) cannot be solved by cycle-consistency checking then we can express a linear equation modulo p p ppp using Γ Î“ Gamma\GammaΓ. Since our constraint language is on a domain A A AAA, we could not expect to pp-define the relation x 1 + x 2 = x 3 + x 4 ( mod p ) x 1 + x 2 = x 3 + x 4 ( mod p ) x_(1)+x_(2)=x_(3)+x_(4)(mod p)x_{1}+x_{2}=x_{3}+x_{4}(\bmod p)x1+x2=x3+x4(modp). Instead, we claim that there exist S A S ⊆ A S sube AS \subseteq AS⊆A and a surjective mapping φ : A Z p s φ : A → Z p s varphi:A rarrZ_(p)^(s)\varphi: A \rightarrow \mathbb{Z}_{p}^{s}φ:A→Zps such that the relation
(2.6) { ( a 1 , a 2 , a 3 , a 4 } a 1 , a 2 , a 3 , a 4 S , φ ( a 1 ) + φ ( a 2 ) = φ ( a 3 ) + φ ( a 4 ) } (2.6) a 1 , a 2 , a 3 , a 4 ∣ a 1 , a 2 , a 3 , a 4 ∈ S , φ a 1 + φ a 2 = φ a 3 + φ a 4 {:(2.6){(a_(1),a_(2),a_(3),a_(4)}∣a_(1),a_(2),a_(3),a_(4)in S,varphi(a_(1))+varphi(a_(2))=varphi(a_(3))+varphi(a_(4))}:}\begin{equation*} \left\{\left(a_{1}, a_{2}, a_{3}, a_{4}\right\} \mid a_{1}, a_{2}, a_{3}, a_{4} \in S, \varphi\left(a_{1}\right)+\varphi\left(a_{2}\right)=\varphi\left(a_{3}\right)+\varphi\left(a_{4}\right)\right\} \tag{2.6} \end{equation*}(2.6){(a1,a2,a3,a4}∣a1,a2,a3,a4∈S,φ(a1)+φ(a2)=φ(a3)+φ(a4)}
is pp-definable. This means that the linear equation is defined on some S S SSS modulo some equivalence relation defined by φ φ varphi\varphiφ. To avoid such a transformation, we could introduce the notion of pp-constructability and say that x 1 + x 2 = x 3 + x 4 ( mod p ) x 1 + x 2 = x 3 + x 4 ( mod p ) x_(1)+x_(2)=x_(3)+x_(4)(mod p)x_{1}+x_{2}=x_{3}+x_{4}(\bmod p)x1+x2=x3+x4(modp) is pp-constructable from Γ Î“ Gamma\GammaΓ. To keep everything simple, we do not define pp-constructability and use it informally hoping that the idea of this notion is clear from our example. For more details about pp-constructability, see [7].
If such a linear equation cannot be pp-defined (pp-constructed) then there should be some operation that preserves Γ Î“ Gamma\GammaΓ but not the linear equation modulo p p ppp. An operation f f fff is called a a aaa Weak Near Unanimity Operation (WNU) if it satisfies the following identity:
f ( y , x , x , , x ) = f ( x , y , x , , x ) = = f ( x , x , , x , y ) f ( y , x , x , … , x ) = f ( x , y , x , … , x ) = ⋯ = f ( x , x , … , x , y ) f(y,x,x,dots,x)=f(x,y,x,dots,x)=cdots=f(x,x,dots,x,y)f(y, x, x, \ldots, x)=f(x, y, x, \ldots, x)=\cdots=f(x, x, \ldots, x, y)f(y,x,x,…,x)=f(x,y,x,…,x)=⋯=f(x,x,…,x,y)
It is not hard to check that an idempotent W N U W N U WNU\mathrm{WNU}WNU of arity p p ppp does not preserve a nontrivial linear equation modulo p p ppp (see Lemma 4.9 in [47]). Thus, the existence of an idempotent p p ppp-ary WNU polymorphism of Γ Î“ Gamma\GammaΓ guarantees that a linear equation modulo p p ppp cannot be p p p p pp\mathrm{pp}pp defined (pp-constructed). That is why a relation satisfying (2.6) is called p p ppp-WNU-blocker Hence, if Γ Î“ Gamma\GammaΓ has WNU polymorphisms of all arities then no linear equations can appear. The following theorem confirms that nothing but linear equations could be an obstacle for the local consistency checking.
Theorem 2.3 ([47]). Suppose Γ Î“ Gamma\GammaΓ is a constraint language containing all constant relations. The following conditions are equivalent:
(1) every cycle-consistent instance of CSP ( Γ ) CSP ⁡ ( Γ ) CSP(Gamma)\operatorname{CSP}(\Gamma)CSP⁡(Γ) has a solution;
(2) Γ Î“ Gamma\GammaΓ has a WNU polymorphisms of all arities n 3 n ≥ 3 n >= 3n \geq 3n≥3;
(3) there does not exist a p p ppp-WNU-blocker pp-definable from Γ Î“ Gamma\GammaΓ.
Thus, the fact that we cannot express (pp-define, pp-construct) a nontrivial linear equation makes the problem solvable by the cycle-consistency checking.

2.7. CSP Dichotomy Conjecture

In this subsection, we formulate a criterion for CSP ( Γ ) CSP ⁡ ( Γ ) CSP(Gamma)\operatorname{CSP}(\Gamma)CSP⁡(Γ) to be solvable in polynomial time. This criterion is known as the CSP Dichotomy Conjecture, it was formulated almost 30 years ago [28,29] but was an open question until 2017 [19,20,42,44].
Theorem 2.4 ( [ 19 , 20 , 42 , 44 ] ) [ 19 , 20 , 42 , 44 ] ) [19,20,42,44])[19,20,42,44])[19,20,42,44]). Suppose Γ Î“ Gamma\GammaΓ is a constraint language on a finite set A A AAA. Then
(1) CSP ( Γ ) CSP ⁡ ( Γ ) CSP(Gamma)\operatorname{CSP}(\Gamma)CSP⁡(Γ) is solvable in polynomial time if Γ Î“ Gamma\GammaΓ is preserved by a W N U W N U WNUW N UWNU;
(2) CSP ( Γ ) CSP ⁡ ( Γ ) CSP(Gamma)\operatorname{CSP}(\Gamma)CSP⁡(Γ) is NP-complete otherwise.
The reason why the existence of a WNU polymorphism makes the problem easy is the fact that we cannot pp-define a strong relation giving us NP-hardness. A relation R = ( B 0 B 1 ) 3 ( B 0 3 B 1 3 ) R = B 0 ∪ B 1 3 ∖ B 0 3 ∪ B 1 3 R=(B_(0)uuB_(1))^(3)\\(B_(0)^(3)uuB_(1)^(3))R=\left(B_{0} \cup B_{1}\right)^{3} \backslash\left(B_{0}^{3} \cup B_{1}^{3}\right)R=(B0∪B1)3∖(B03∪B13), where B 0 , B 1 A , B 0 , B 1 B 0 , B 1 ⊆ A , B 0 ≠ ∅ , B 1 ≠ ∅ B_(0),B_(1)sube A,B_(0)!=O/,B_(1)!=O/B_{0}, B_{1} \subseteq A, B_{0} \neq \varnothing, B_{1} \neq \varnothingB0,B1⊆A,B0≠∅,B1≠∅, and B 0 B 1 = B 0 ∩ B 1 = ∅ B_(0)nnB_(1)=O/B_{0} \cap B_{1}=\varnothingB0∩B1=∅, is called a WNU-blocker. Such relations are similar to the not-all-equal (NAE) relation on { 0 , 1 } { 0 , 1 } {0,1}\{0,1\}{0,1}, where B 0 B 0 B_(0)B_{0}B0 means 0 and B 1 B 1 B_(1)B_{1}B1 means 1 . Instead of the existence of a pp-definable WNU-blocker, we could say that the relation NAE is pp-constructable from Γ Î“ Gamma\GammaΓ. Note that CSP ( { N A E } ) CSP ⁡ ( { N A E } ) CSP({NAE})\operatorname{CSP}(\{\mathrm{NAE}\})CSP⁡({NAE}) and CSP ( { R } ) CSP ⁡ ( { R } ) CSP({R})\operatorname{CSP}(\{R\})CSP⁡({R}) for a WNU-blocker R R RRR are NP-complete problems.
We can check (see Lemma 4.8 in [47]) that a WNU operation does not preserve a WNU-blocker. Moreover, we have the following criterion.
Lemma 2.5 ([47]). A constraint language Γ Î“ Gamma\GammaΓ containing all constant relations is preserved by a WNU if and only if there is no WNU-blocker pp-definable from Γ Î“ Gamma\GammaΓ.
Thus, CSP ( Γ ) CSP ⁡ ( Γ ) CSP(Gamma)\operatorname{CSP}(\Gamma)CSP⁡(Γ) is solvable in polynomial time if and only if a WNU-blocker cannot be pp-defined. Hence, the fact that we cannot pp-construct the not-all-equal relation makes the problem easy, and a W N U W N U WNU\mathrm{WNU}WNU is an operation that guarantees that this relation cannot be pp-constructed.

2.8. How to solve CSP if pp-definable relations are simple

Below we discuss how the fact that only simple relations can be pp-defined from Γ Î“ Gamma\GammaΓ can help to solve CSP ( Γ ) CSP ⁡ ( Γ ) CSP(Gamma)\operatorname{CSP}(\Gamma)CSP⁡(Γ) in polynomial time. In this case we can calculate the sentence explicitly eliminating existential quantifiers one by one. I believe that a similar idea should work for any Γ Î“ Gamma\GammaΓ preserved by a WNU, which will give us a simple algorithm for CSP ( Γ ) CSP ⁡ ( Γ ) CSP(Gamma)\operatorname{CSP}(\Gamma)CSP⁡(Γ).
CSP ( Γ ) CSP ⁡ ( Γ ) CSP(Gamma)\operatorname{CSP}(\Gamma)CSP⁡(Γ) can be viewed as the following problem. Given a sentence
x 1 x n ( R 1 ( v 1 , 1 , , v 1 , n 1 ) R s ( v s , 1 , , v s , n s ) ) ∃ x 1 … ∃ x n R 1 v 1 , 1 , … , v 1 , n 1 ∧ ⋯ ∧ R s v s , 1 , … , v s , n s EEx_(1)dots EEx_(n)(R_(1)(v_(1,1),dots,v_(1,n_(1)))^^cdots^^R_(s)(v_(s,1),dots,v_(s,n_(s))))\exists x_{1} \ldots \exists x_{n}\left(R_{1}\left(v_{1,1}, \ldots, v_{1, n_{1}}\right) \wedge \cdots \wedge R_{s}\left(v_{s, 1}, \ldots, v_{s, n_{s}}\right)\right)∃x1…∃xn(R1(v1,1,…,v1,n1)∧⋯∧Rs(vs,1,…,vs,ns))
we need to check whether it holds. To do this, let us remove the quantifiers one by one. Let
Δ n 1 ( x 1 , , x n 1 ) = x n ( R 1 ( v 1 , 1 , , v 1 , n 1 ) R s ( v s , 1 , , v s , n s ) ) Δ n − 1 x 1 , … , x n − 1 = ∃ x n R 1 v 1 , 1 , … , v 1 , n 1 ∧ ⋯ ∧ R s v s , 1 , … , v s , n s Delta_(n-1)(x_(1),dots,x_(n-1))=EEx_(n)(R_(1)(v_(1,1),dots,v_(1,n_(1)))^^cdots^^R_(s)(v_(s,1),dots,v_(s,n_(s))))\Delta_{n-1}\left(x_{1}, \ldots, x_{n-1}\right)=\exists x_{n}\left(R_{1}\left(v_{1,1}, \ldots, v_{1, n_{1}}\right) \wedge \cdots \wedge R_{s}\left(v_{s, 1}, \ldots, v_{s, n_{s}}\right)\right)Δn−1(x1,…,xn−1)=∃xn(R1(v1,1,…,v1,n1)∧⋯∧Rs(vs,1,…,vs,ns))
In general, Δ n 1 Δ n − 1 Delta_(n-1)\Delta_{n-1}Δn−1 could be any relation of arity n 1 n − 1 n-1n-1n−1, and even to write this relation we need | A | n 1 | A | n − 1 |A|^(n-1)|A|^{n-1}|A|n−1 bits. Nevertheless, we believe that if CSP ( Γ ) CSP ⁡ ( Γ ) CSP(Gamma)\operatorname{CSP}(\Gamma)CSP⁡(Γ) is tractable then the relation Δ n 1 Δ n − 1 Delta_(n-1)\Delta_{n-1}Δn−1 (or
the important part of it) has a compact representation and can be efficiently computed. Then we calculate Δ n 2 , Δ n 3 , , Δ 0 Δ n − 2 , Δ n − 3 , … , Δ 0 Delta_(n-2),Delta_(n-3),dots,Delta_(0)\Delta_{n-2}, \Delta_{n-3}, \ldots, \Delta_{0}Δn−2,Δn−3,…,Δ0, where Δ i 1 ( x 1 , , x i 1 ) = x i Δ i ( x 1 , , x i ) Δ i − 1 x 1 , … , x i − 1 = ∃ x i Δ i x 1 , … , x i Delta_(i-1)(x_(1),dots,x_(i-1))=EEx_(i)Delta_(i)(x_(1),dots,x_(i))\Delta_{i-1}\left(x_{1}, \ldots, x_{i-1}\right)=\exists x_{i} \Delta_{i}\left(x_{1}, \ldots, x_{i}\right)Δi−1(x1,…,xi−1)=∃xiΔi(x1,…,xi), and the value of Δ 0 Δ 0 Delta_(0)\Delta_{0}Δ0 is the answer we need.
We may check that on a 2-element domain we have
x n ( R 1 ( v 1 , 1 , , v 1 , n 1 ) R s ( v s , 1 , , v s , n s ) ) (2.7) = i , j { 1 , 2 , , s } ( x n R i ( v i , 1 , , v i , n i ) R j ( v j , 1 , , v j , n j ) ) ∃ x n R 1 v 1 , 1 , … , v 1 , n 1 ∧ ⋯ ∧ R s v s , 1 , … , v s , n s (2.7) = ⋀ i , j ∈ { 1 , 2 , … , s }   ∃ x n R i v i , 1 , … , v i , n i ∧ R j v j , 1 , … , v j , n j {:[EEx_(n)(R_(1)(v_(1,1),dots,v_(1,n_(1)))^^cdots^^R_(s)(v_(s,1),dots,v_(s,n_(s))))],[(2.7)quad=^^^_(i,j in{1,2,dots,s})(EEx_(n)R_(i)(v_(i,1),dots,v_(i,n_(i)))^^R_(j)(v_(j,1),dots,v_(j,n_(j))))]:}\begin{align*} & \exists x_{n}\left(R_{1}\left(v_{1,1}, \ldots, v_{1, n_{1}}\right) \wedge \cdots \wedge R_{s}\left(v_{s, 1}, \ldots, v_{s, n_{s}}\right)\right) \\ & \quad=\bigwedge_{i, j \in\{1,2, \ldots, s\}}\left(\exists x_{n} R_{i}\left(v_{i, 1}, \ldots, v_{i, n_{i}}\right) \wedge R_{j}\left(v_{j, 1}, \ldots, v_{j, n_{j}}\right)\right) \tag{2.7} \end{align*}∃xn(R1(v1,1,…,v1,n1)∧⋯∧Rs(vs,1,…,vs,ns))(2.7)=⋀i,j∈{1,2,…,s}(∃xnRi(vi,1,…,vi,ni)∧Rj(vj,1,…,vj,nj))
The implication ⇒ =>\Rightarrow⇒ is obvious. To prove ⇐ lArr\Leftarrow⇐ assume that the left-hand side does not hold. Then the conjunctive part does not hold on both ( x 1 , , x n 1 , 0 ) x 1 , … , x n − 1 , 0 (x_(1),dots,x_(n-1),0)\left(x_{1}, \ldots, x_{n-1}, 0\right)(x1,…,xn−1,0) and ( x 1 , , x n 1 , 1 ) x 1 , … , x n − 1 , 1 (x_(1),dots,x_(n-1),1)\left(x_{1}, \ldots, x_{n-1}, 1\right)(x1,…,xn−1,1). Hence, there exist i i iii and j j jjj such that R i ( v i , 1 , , v i , n i ) R i v i , 1 , … , v i , n i R_(i)(v_(i,1),dots,v_(i,n_(i)))R_{i}\left(v_{i, 1}, \ldots, v_{i, n_{i}}\right)Ri(vi,1,…,vi,ni) does not hold on ( x 1 , x 2 , , x n 1 , 0 ) x 1 , x 2 , … , x n − 1 , 0 (x_(1),x_(2),dots,x_(n-1),0)\left(x_{1}, x_{2}, \ldots, x_{n-1}, 0\right)(x1,x2,…,xn−1,0) and R j ( v j , 1 , , v j , n j ) R j v j , 1 , … , v j , n j R_(j)(v_(j,1),dots,v_(j,n_(j)))R_{j}\left(v_{j, 1}, \ldots, v_{j, n_{j}}\right)Rj(vj,1,…,vj,nj) does not hold on ( x 1 , x 2 , , x n 1 , 1 ) x 1 , x 2 , … , x n − 1 , 1 (x_(1),x_(2),dots,x_(n-1),1)\left(x_{1}, x_{2}, \ldots, x_{n-1}, 1\right)(x1,x2,…,xn−1,1). Hence, the ( i , j ) ( i , j ) (i,j)(i, j)(i,j)-part of the righthand side does not hold.
There are two problems if we use (2.7) to solve the CSP. First, as we mentioned above, the relation R i , j ( ) = x n R i ( v i , 1 , , v i , n i ) R j ( v j , 1 , , v j , n j ) R i , j ( … ) = ∃ x n R i v i , 1 , … , v i , n i ∧ R j v j , 1 , … , v j , n j R_(i,j)(dots)=EEx_(n)R_(i)(v_(i,1),dots,v_(i,n_(i)))^^R_(j)(v_(j,1),dots,v_(j,n_(j)))R_{i, j}(\ldots)=\exists x_{n} R_{i}\left(v_{i, 1}, \ldots, v_{i, n_{i}}\right) \wedge R_{j}\left(v_{j, 1}, \ldots, v_{j, n_{j}}\right)Ri,j(…)=∃xnRi(vi,1,…,vi,ni)∧Rj(vj,1,…,vj,nj) probably does not have a compact representation. Second, if we remove the quantifiers x n , x n 1 , , x 1 ∃ x n , ∃ x n − 1 , … , ∃ x 1 EEx_(n),EEx_(n-1),dots,EEx_(1)\exists x_{n}, \exists x_{n-1}, \ldots, \exists x_{1}∃xn,∃xn−1,…,∃x1 one by one, potentially we could get an exponential number of relations in the formula. Let us show how these problem are solved for concrete examples on a 2-element domain.

2.9. System of linear equations in Z 2 Z 2 Z_(2)\mathbb{Z}_{2}Z2

Let A = { 0 , 1 } A = { 0 , 1 } A={0,1}A=\{0,1\}A={0,1} and let Γ Î“ Gamma\GammaΓ consist of linear equations in Z 2 Z 2 Z_(2)\mathbb{Z}_{2}Z2. Suppose that for every i i iii we have
R i ( v i , 1 , , v i , n i ) = ( a 1 i x n + a 2 i x 2 + + a n i x n = a 0 i ) R i v i , 1 , … , v i , n i = a 1 i x n + a 2 i x 2 + ⋯ + a n i x n = a 0 i R_(i)(v_(i,1),dots,v_(i,n_(i)))=(a_(1)^(i)x_(n)+a_(2)^(i)x_(2)+cdots+a_(n)^(i)x_(n)=a_(0)^(i))R_{i}\left(v_{i, 1}, \ldots, v_{i, n_{i}}\right)=\left(a_{1}^{i} x_{n}+a_{2}^{i} x_{2}+\cdots+a_{n}^{i} x_{n}=a_{0}^{i}\right)Ri(vi,1,…,vi,ni)=(a1ixn+a2ix2+⋯+anixn=a0i)
For a n i = a n j = 1 a n i = a n j = 1 a_(n)^(i)=a_(n)^(j)=1a_{n}^{i}=a_{n}^{j}=1ani=anj=1, we have
R i , j ( ) := x n ( R i ( v i , 1 , , v i , n i ) R j ( v j , 1 , , v j , n j ) ) = ( a 1 i x 1 + a 2 i x 2 + + a n 1 i x n 1 + a 0 i = a 1 j x 1 + a 2 j x 2 + + a n 1 j x n 1 + a 0 j ) R i , j ( … ) := ∃ x n R i v i , 1 , … , v i , n i ∧ R j v j , 1 , … , v j , n j = a 1 i x 1 + a 2 i x 2 + ⋯ + a n − 1 i x n − 1 + a 0 i = a 1 j x 1 + a 2 j x 2 + ⋯ + a n − 1 j x n − 1 + a 0 j {:[R_(i,j)(dots)],[quad:=EEx_(n)(R_(i)(v_(i,1),dots,v_(i,n_(i)))^^R_(j)(v_(j,1),dots,v_(j,n_(j))))],[quad=(a_(1)^(i)x_(1)+a_(2)^(i)x_(2)+cdots+a_(n-1)^(i)x_(n-1)+a_(0)^(i)=a_(1)^(j)x_(1)+a_(2)^(j)x_(2)+cdots+a_(n-1)^(j)x_(n-1)+a_(0)^(j))]:}\begin{aligned} & R_{i, j}(\ldots) \\ & \quad:=\exists x_{n}\left(R_{i}\left(v_{i, 1}, \ldots, v_{i, n_{i}}\right) \wedge R_{j}\left(v_{j, 1}, \ldots, v_{j, n_{j}}\right)\right) \\ & \quad=\left(a_{1}^{i} x_{1}+a_{2}^{i} x_{2}+\cdots+a_{n-1}^{i} x_{n-1}+a_{0}^{i}=a_{1}^{j} x_{1}+a_{2}^{j} x_{2}+\cdots+a_{n-1}^{j} x_{n-1}+a_{0}^{j}\right) \end{aligned}Ri,j(…):=∃xn(Ri(vi,1,…,vi,ni)∧Rj(vj,1,…,vj,nj))=(a1ix1+a2ix2+⋯+an−1ixn−1+a0i=a1jx1+a2jx2+⋯+an−1jxn−1+a0j)
If a n i = 0 a n i = 0 a_(n)^(i)=0a_{n}^{i}=0ani=0 then the constraint R i ( v i , 1 , , v i , n i ) R i v i , 1 , … , v i , n i R_(i)(v_(i,1),dots,v_(i,n_(i)))R_{i}\left(v_{i, 1}, \ldots, v_{i, n_{i}}\right)Ri(vi,1,…,vi,ni) does not depend on x n x n x_(n)x_{n}xn, so we keep it as it is when remove the quantifier. Hence, in every case we have a compact representation of Δ n 1 Δ n − 1 Delta_(n-1)\Delta_{n-1}Δn−1. To avoid the exponential growth of the number of the constraints, we use the idea from the Gaussian elimination. Choose k k kkk such that a n k = 1 a n k = 1 a_(n)^(k)=1a_{n}^{k}=1ank=1, then calculate only R k , 1 , , R k , s R k , 1 , … , R k , s R_(k,1),dots,R_(k,s)R_{k, 1}, \ldots, R_{k, s}Rk,1,…,Rk,s and ignore all the other relations. Thus, in this case we have
Δ n 1 ( x 1 , , x n 1 ) = x n ( R 1 ( v 1 , 1 , , v 1 , n 1 ) R s ( v s , 1 , , v s , n s ) ) = j { 1 , 2 , , s } ( x n R k ( v k , 1 , , v k , n k ) R j ( v j , 1 , , v j , n j ) ) Δ n − 1 x 1 , … , x n − 1 = ∃ x n R 1 v 1 , 1 , … , v 1 , n 1 ∧ ⋯ ∧ R s v s , 1 , … , v s , n s = â‹€ j ∈ { 1 , 2 , … , s }   ∃ x n R k v k , 1 , … , v k , n k ∧ R j v j , 1 , … , v j , n j {:[Delta_(n-1)(x_(1),dots,x_(n-1))=EEx_(n)(R_(1)(v_(1,1),dots,v_(1,n_(1)))^^cdots^^R_(s)(v_(s,1),dots,v_(s,n_(s))))],[=^^^_(j in{1,2,dots,s})(EEx_(n)R_(k)(v_(k,1),dots,v_(k,n_(k)))^^R_(j)(v_(j,1),dots,v_(j,n_(j))))]:}\begin{align*} \Delta_{n-1}\left(x_{1}, \ldots, x_{n-1}\right) & =\exists x_{n}\left(R_{1}\left(v_{1,1}, \ldots, v_{1, n_{1}}\right) \wedge \cdots \wedge R_{s}\left(v_{s, 1}, \ldots, v_{s, n_{s}}\right)\right) \\ & =\bigwedge_{j \in\{1,2, \ldots, s\}}\left(\exists x_{n} R_{k}\left(v_{k, 1}, \ldots, v_{k, n_{k}}\right) \wedge R_{j}\left(v_{j, 1}, \ldots, v_{j, n_{j}}\right)\right) \end{align*}Δn−1(x1,…,xn−1)=∃xn(R1(v1,1,…,v1,n1)∧⋯∧Rs(vs,1,…,vs,ns))=â‹€j∈{1,2,…,s}(∃xnRk(vk,1,…,vk,nk)∧Rj(vj,1,…,vj,nj))
Proceeding this way, we calculate Δ n 2 , Δ n 3 , , Δ 0 Δ n − 2 , Δ n − 3 , … , Δ 0 Delta_(n-2),Delta_(n-3),dots,Delta_(0)\Delta_{n-2}, \Delta_{n-3}, \ldots, \Delta_{0}Δn−2,Δn−3,…,Δ0. Note that (2.8) holds not only for linear equations but whenever a variable x n x n x_(n)x_{n}xn is uniquely determined by the other variables in R k ( v k , 1 , , v k , n 1 ) R k v k , 1 , … , v k , n 1 R_(k)(v_(k,1),dots,v_(k,n_(1)))R_{k}\left(v_{k, 1}, \ldots, v_{k, n_{1}}\right)Rk(vk,1,…,vk,n1).

2.10. 2-satisfability

Let A = { 0 , 1 } A = { 0 , 1 } A={0,1}A=\{0,1\}A={0,1} and let Γ Î“ Gamma\GammaΓ consist of all binary relations. In this case R i , j R i , j R_(i,j)R_{i, j}Ri,j is also binary, which means that we do not have a problem with a compact representation. Also, every time we eliminate a quantifier and caclulate Δ i Δ i Delta_(i)\Delta_{i}Δi, we remove the repetitive constraints. Therefore, in each Δ i Δ i Delta_(i)\Delta_{i}Δi we cannot have more than i i 2 2 2 i â‹… i â‹… 2 2 2 i*i*2^(2^(2))i \cdot i \cdot 2^{2^{2}}iâ‹…iâ‹…222 constraints because we have i i iii different variables and 2 2 2 2 2 2 2^(2^(2))2^{2^{2}}222 different binary relations on { 0 , 1 } { 0 , 1 } {0,1}\{0,1\}{0,1}.
As we see, the main question in both examples is the existence of a compact representation. In the first example we represent any relation as a conjunction of linear equations, in the second we represent as a conjunction of binary relations. We could ask when such a compact representation exists. Let s Γ ( n ) s Γ ( n ) s_(Gamma)(n)s_{\Gamma}(n)sΓ(n) be the number of pp-definable from Γ Î“ Gamma\GammaΓ relations of arity n n nnn. If log 2 s Γ ( n ) log 2 ⁡ s Γ ( n ) log_(2)s_(Gamma)(n)\log _{2} s_{\Gamma}(n)log2⁡sΓ(n) grows exponentially then we need exponential space to encode relations of arity n n nnn and we cannot expect a compact representation. We say that Γ Î“ Gamma\GammaΓ has few subpowers if log 2 s Γ ( n ) < p ( n ) log 2 ⁡ s Γ ( n ) < p ( n ) log_(2)s_(Gamma)(n) < p(n)\log _{2} s_{\Gamma}(n)<p(n)log2⁡sΓ(n)<p(n) for a polynomial p ( n ) p ( n ) p(n)p(n)p(n). It turns out that there is a simple criterion for the constraint language to have few subpowers. An operation t t ttt is called an edge operation if it satisfies the following identities:
t ( x , x , y , y , y , , y , y ) = y t ( x , y , x , y , y , , y , y ) = y t ( y , y , y , x , y , , y , y ) = y t ( y , y , y , y , x , , y , y ) = y t ( y , y , y , y , y , , x , y ) = y t ( y , y , y , y , y , , y , x ) = y t ( x , x , y , y , y , … , y , y ) = y t ( x , y , x , y , y , … , y , y ) = y t ( y , y , y , x , y , … , y , y ) = y t ( y , y , y , y , x , … , y , y ) = y … t ( y , y , y , y , y , … , x , y ) = y t ( y , y , y , y , y , … , y , x ) = y {:[t(x","x","y","y","y","dots","y","y)=y],[t(x","y","x","y","y","dots","y","y)=y],[t(y","y","y","x","y","dots","y","y)=y],[t(y","y","y","y","x","dots","y","y)=y],[dots],[t(y","y","y","y","y","dots","x","y)=y],[t(y","y","y","y","y","dots","y","x)=y]:}\begin{gathered} t(x, x, y, y, y, \ldots, y, y)=y \\ t(x, y, x, y, y, \ldots, y, y)=y \\ t(y, y, y, x, y, \ldots, y, y)=y \\ t(y, y, y, y, x, \ldots, y, y)=y \\ \ldots \\ t(y, y, y, y, y, \ldots, x, y)=y \\ t(y, y, y, y, y, \ldots, y, x)=y \end{gathered}t(x,x,y,y,y,…,y,y)=yt(x,y,x,y,y,…,y,y)=yt(y,y,y,x,y,…,y,y)=yt(y,y,y,y,x,…,y,y)=y…t(y,y,y,y,y,…,x,y)=yt(y,y,y,y,y,…,y,x)=y
Theorem 2.6 ([9]). A constraint language Γ Î“ Gamma\GammaΓ containing all constant relations has few subpowers if and only if it has an edge polymorphism.
We can show that if Γ Î“ Gamma\GammaΓ has few subpowers then the pp-definable relations have a natural compact representation, which gives a polynomial algorithm for CSP ( Г ) CSP ⁡ ( Г ) CSP(Г)\operatorname{CSP}(Г)CSP⁡(Г) [33]. Note that two examples of an edge operation were given earlier in this paper. The first example is a majority operation satisfying m ( y , y , x ) = m ( y , x , y ) = m ( x , y , y ) = y m ( y , y , x ) = m ( y , x , y ) = m ( x , y , y ) = y m(y,y,x)=m(y,x,y)=m(x,y,y)=ym(y, y, x)=m(y, x, y)=m(x, y, y)=ym(y,y,x)=m(y,x,y)=m(x,y,y)=y. By adding 3 dummy variables in the beginning, we get the required properties of an edge operation. Another example is x + y + z x + y + z x+y+zx+y+zx+y+z on { 0 , 1 } { 0 , 1 } {0,1}\{0,1\}{0,1}. By adding dummy variables at the end, we can easily satisfy all the identities. Very roughly speaking, any few subpowers case is just a combination (probably very complicated) of the majority case and the linear case.

2.11. Strong subuniverses and a proof of the CSP Dichotomy Conjecture

In this subsection, we consider another simple idea that can solve the CSP in polynomial time. This idea is one of the two main ingredients of the proof of the CSP Dichotomy Conjecture in [ 42 , 44 ] [ 42 , 44 ] [42,44][42,44][42,44].
Assume that for every variable x x xxx whose domain is D x , | D x | > 1 D x , D x > 1 D_(x),|D_(x)| > 1D_{x},\left|D_{x}\right|>1Dx,|Dx|>1, we can choose a subset B x D x B x ⊊ D x B_(x)⊊D_(x)B_{x} \subsetneq D_{x}Bx⊊Dx such that if the instance has a solution, then it has a solution with x B x x ∈ B x x inB_(x)x \in B_{x}x∈Bx.
In this case we can reduce the domains iteratively until the moment when each domain has exactly one element, which usually gives us a solution.
As we saw in Section 2.5, if Γ Î“ Gamma\GammaΓ is preserved by x y x ∨ y x vv yx \vee yx∨y and the instance is 1-consistent then we can safely reduce the domain of a variable to { 1 } { 1 } {1}\{1\}{1}. Similarly, if Γ Î“ Gamma\GammaΓ is preserved by the majority operation x y y z x z x y ∨ y z ∨ x z xy vv yz vv xzx y \vee y z \vee x zxy∨yz∨xz and the instance is cycle-consistent, then we can safely reduce the domain { 0 , 1 } { 0 , 1 } {0,1}\{0,1\}{0,1} to { 0 } { 0 } {0}\{0\}{0} and { 1 } { 1 } {1}\{1\}{1} [47]. It turns out that this idea can be generalized for any constraint language preserved by a WNU operation.
A unary relation B A B ⊆ A B sube AB \subseteq AB⊆A is called a subuniverse if B B BBB is pp-definable over Γ Î“ Gamma\GammaΓ. It can be easily checked that all the domains D x D x D_(x)D_{x}Dx that appear while checking consistency (see Section 2.4) are subuniverses. Let us define three types of strong subuniverses:
Binary absorbing subuniverse. We say that B B ′ B^(')B^{\prime}B′ is a binary absorbing subuniverse of B B BBB if there exists a binary operation f Pol ( Γ ) f ∈ Pol ⁡ ( Γ ) f in Pol(Gamma)f \in \operatorname{Pol}(\Gamma)f∈Pol⁡(Γ) such that f ( B , B ) B f B ′ , B ⊆ B ′ f(B^('),B)subeB^(')f\left(B^{\prime}, B\right) \subseteq B^{\prime}f(B′,B)⊆B′ and f ( B , B ) B f B , B ′ ⊆ B ′ f(B,B^('))subeB^(')f\left(B, B^{\prime}\right) \subseteq B^{\prime}f(B,B′)⊆B′. For example, if the operation x y x ∨ y x vv yx \vee yx∨y preserves Γ Î“ Gamma\GammaΓ then { 1 } { 1 } {1}\{1\}{1} is a binary absorbing subuniverse of { 0 , 1 } { 0 , 1 } {0,1}\{0,1\}{0,1} and x y x ∨ y x vv yx \vee yx∨y is a binary absorbing operation.
Ternary absorbing subuniverse. We say that B B ′ B^(')B^{\prime}B′ is a ternary absorbing subuniverse of B B BBB if there exists a ternary operation f Pol ( Γ ) f ∈ Pol ⁡ ( Γ ) f in Pol(Gamma)f \in \operatorname{Pol}(\Gamma)f∈Pol⁡(Γ) such that f ( B , B , B ) B , f ( B , B , B ) B f B ′ , B ′ , B ⊆ B ′ , f B ′ , B , B ′ ⊆ B ′ f(B^('),B^('),B)subeB^('),f(B^('),B,B^('))subeB^(')f\left(B^{\prime}, B^{\prime}, B\right) \subseteq B^{\prime}, f\left(B^{\prime}, B, B^{\prime}\right) \subseteq B^{\prime}f(B′,B′,B)⊆B′,f(B′,B,B′)⊆B′, and f ( B , B , B ) B f B , B ′ , B ′ ⊆ B ′ f(B,B^('),B^('))subeB^(')f\left(B, B^{\prime}, B^{\prime}\right) \subseteq B^{\prime}f(B,B′,B′)⊆B′. For example, if the majority operation x y y z x z x y ∨ y z ∨ x z xy vv yz vv xzx y \vee y z \vee x zxy∨yz∨xz preserves Γ Î“ Gamma\GammaΓ, then both { 0 } { 0 } {0}\{0\}{0} and { 1 } { 1 } {1}\{1\}{1} are ternary absorbing subuniverses of { 0 , 1 } { 0 , 1 } {0,1}\{0,1\}{0,1}. Since we can always add a dummy variable to a binary absorbing operation, any binary absorbing subuniverse is also a ternary absorbing subuniverse.
To define the last type of strong subalgebras we need some understanding of the Universal Algebra. We do not think a concrete definition is important here, that is why if a reader thinks the definition is too complicated, we recommend to skip it and think about the last type as something similar to the first two.
PC subuniverse. A set F F FFF of operations is called Polynomially Complete (PC) if any operation can be derived from F F FFF and constants using composition. We say that B B ′ B^(')B^{\prime}B′ is a PC subuniverse of B B BBB if there exists a pp-definable equivalence relation σ B × B σ ⊆ B × B sigma sube B xx B\sigma \subseteq B \times Bσ⊆B×B such that Pol ( Γ ) / σ Pol ⁡ ( Γ ) / σ Pol(Gamma)//sigma\operatorname{Pol}(\Gamma) / \sigmaPol⁡(Γ)/σ is P C P C PC\mathrm{PC}PC.
A subset B B ′ B^(')B^{\prime}B′ of B B BBB is called a strong subuniverse if B B ′ B^(')B^{\prime}B′ is a ternary absorbing subuniverse or a PC subuniverse.
Theorem 2.7 ([47]). Suppose Γ Î“ Gamma\GammaΓ contains all constant relations and is preserved by a WNU operation, B A B ⊆ A B sube AB \subseteq AB⊆A is a subuniverse. Then
(1) there exists a strong subuniverse B B B ′ ⊊ B B^(')⊊BB^{\prime} \subsetneq BB′⊊B, or
(2) there exists a pp-definable nontrivial equivalence relation σ σ sigma\sigmaσ on B B BBB and f Pol ( Γ ) f ∈ Pol ⁡ ( Γ ) f in Pol(Gamma)f \in \operatorname{Pol}(\Gamma)f∈Pol⁡(Γ) such that ( B ; f ) / σ ( Z p k ; x y + z ) ( B ; f ) / σ ≅ Z p k ; x − y + z (B;f)//sigma~=(Z_(p)^(k);x-y+z)(B ; f) / \sigma \cong\left(\mathbb{Z}_{p}^{k} ; x-y+z\right)(B;f)/σ≅(Zpk;x−y+z).
As it follows from the next lemma, the second condition implies that any ppdefinable relation (modulo σ σ sigma\sigmaσ ) can be viewed as a system of linear equations in a field.
Lemma 2.8 ([32]). Suppose R Z p n R ⊆ Z p n R subeZ_(p)^(n)R \subseteq \mathbb{Z}_{p}^{n}R⊆Zpn preserved by x y + z x − y + z x-y+zx-y+zx−y+z. Then R R RRR can be represented as a conjunction of relations of the form a 1 x 1 + + a n x n = a 0 ( mod p ) a 1 x 1 + ⋯ + a n x n = a 0 ( mod p ) a_(1)x_(1)+cdots+a_(n)x_(n)=a_(0)(mod p)a_{1} x_{1}+\cdots+a_{n} x_{n}=a_{0}(\bmod p)a1x1+⋯+anxn=a0(modp).
For CSPs solvable by the local consistency checking, strong subuniverses have the following property.

Theorem 2.9 ([47]). Suppose

(1) Γ Î“ Gamma\GammaΓ is a constraint language containing all constant relations;
(2) Γ Î“ Gamma\GammaΓ is preserved by a W N U W N U WNUW N UWNU of each arity n 3 n ≥ 3 n >= 3n \geq 3n≥3;
(3) d d ddd is a cycle-consistent instance of CSP ( Γ ) CSP ⁡ ( Γ ) CSP(Gamma)\operatorname{CSP}(\Gamma)CSP⁡(Γ);
(4) D x D x D_(x)D_{x}Dx is the domain of a variable x x xxx;
(5) B B BBB is a strong subalgebra of D x D x D_(x)D_{x}Dx.
Then & has a solution with x B x ∈ B x in Bx \in Bx∈B.
Thus, strong subuniverses have the required property that we cannot loose all the solutions when we restrict a variable to it. As it was proved in [44], a similar theorem holds for any constraint language preserved by a WNU operation (with additional consistency conditions on the instance). We skip this result because it would require too many additional definitions.
As we see from Theorem 2.7, for every domain D x D x D_(x)D_{x}Dx either we have a strong subuniverse and can reduce the domain of some variable, or, modulo some equivalence relation, we have a system of linear equations in a field. If Γ Î“ Gamma\GammaΓ has a WNU polymorphism of each arity n 3 n ≥ 3 n >= 3n \geq 3n≥3, then we always have the first case; hence, we can iteratively reduce the domains until the moment when all the domains have just one element, which gives us a solution. That is why any cycle-consistent instance in this situation has a solution. If we always have the second case then this situation is similar to a system of linear equations, but different linear equations can be mixed which makes it impossible to apply usual Gaussian elimination. Nevertheless, the few subpowers algorithm solves the problem [33].
For many years the main obstacle was that these two situations can be mixed and at the moment we do not know an elegant way how to split them. Nevertheless, the general algorithm for tractable CSP presented in [44] is just a smart combination of these two ideas:
  • if there exists a strong subalgebra, reduce
  • if there exists a system of linear equations, solve it.
For more information about this approach as well as its connection with the second general algorithm see [ 3 ] [ 3 ] [3][3][3].

2.12. Conclusions

Even though we still do not have a simple algorithm that solves all tractable Constraint Satisfaction Problems, we understand what makes the problem hard, and what makes
the problem easy. First, we know that in all the hard cases we can pp-construct (pp-define) the not-all-equal relation, which means that all the NP-hard cases have the same nature. Second, if the CSP is not solvable locally then we can pp-construct (pp-define) a linear equation in a field. Moreover, any domain of a tractable CSP either has a strong subalgebra and we can (almost) safely reduce the domain, or there exists a system of linear equations on this domain. This implies that any tractable CSP can be solved by a smart combination of the Gaussian elimination and local consistency checking, and emphasizes the exclusive role of the linear case in Universal Algebra and Computational Complexity.
Note that both CSP algorithms in [20,44] depend exponentially on the size of the domain, and we could ask whether there exists a universal polynomial algorithm that works for any constraint language Γ Î“ Gamma\GammaΓ admitting a WNU polymorphism.
Problem 1. Does there exist a polynomial algorithm for the following decision problem: given a conjunctive formula R 1 ( v 1 , 1 , , v 1 , n 1 ) R s ( v s , 1 , , v s , n s ) R 1 v 1 , 1 , … , v 1 , n 1 ∧ ⋯ ∧ R s v s , 1 , … , v s , n s R_(1)(v_(1,1),dots,v_(1,n_(1)))^^cdots^^R_(s)(v_(s,1),dots,v_(s,n_(s)))R_{1}\left(v_{1,1}, \ldots, v_{1, n_{1}}\right) \wedge \cdots \wedge R_{s}\left(v_{s, 1}, \ldots, v_{s, n_{s}}\right)R1(v1,1,…,v1,n1)∧⋯∧Rs(vs,1,…,vs,ns), where all relations R 1 , , R s R 1 , … , R s R_(1),dots,R_(s)R_{1}, \ldots, R_{s}R1,…,Rs are preserved by a WNU, decide whether this formula is satisfiable.
If the domain is fixed then the above problem can be solved by the algorithms from [19,42]. In fact, we know from [4, THEOREM 4.2] that from a WNU on a domain of size k k kkk we can always derive a WNU (and also a cyclic operation) of any prime arity greater than k k kkk. Thus, we can find finitely many WNU operations on a domain of size k k kkk such that any constraint language preserved by a WNU is preserved by one of them. It remains to apply the algorithm for each WNU and return a solution if one of them gave a solution.

3. QUANTIFIED CSP

A natural generalization of the CSP is the Quantified Constraint Satisfaction Problem (QCSP), where we allow to use both existential and universal quantifiers. Formally, for a constraint language Γ , QCSP ( Γ ) Γ , QCSP ⁡ ( Γ ) Gamma,QCSP(Gamma)\Gamma, \operatorname{QCSP}(\Gamma)Γ,QCSP⁡(Γ) is the problem to evaluate a sentence of the form
x 1 y 1 x n y n R 1 ( v 1 , 1 , , v 1 , n 1 ) R s ( v s , 1 , , v s , n s ) ∀ x 1 ∃ y 1 … ∀ x n ∃ y n R 1 v 1 , 1 , … , v 1 , n 1 ∧ ⋯ ∧ R s v s , 1 , … , v s , n s AAx_(1)EEy_(1)dots AAx_(n)EEy_(n)R_(1)(v_(1,1),dots,v_(1,n_(1)))^^cdots^^R_(s)(v_(s,1),dots,v_(s,n_(s)))\forall x_{1} \exists y_{1} \ldots \forall x_{n} \exists y_{n} R_{1}\left(v_{1,1}, \ldots, v_{1, n_{1}}\right) \wedge \cdots \wedge R_{s}\left(v_{s, 1}, \ldots, v_{s, n_{s}}\right)∀x1∃y1…∀xn∃ynR1(v1,1,…,v1,n1)∧⋯∧Rs(vs,1,…,vs,ns)
where R 1 , , R s Γ R 1 , … , R s ∈ Γ R_(1),dots,R_(s)in GammaR_{1}, \ldots, R_{s} \in \GammaR1,…,Rs∈Γ, and v i , j { x 1 , , x n , y 1 , , y n } v i , j ∈ x 1 , … , x n , y 1 , … , y n v_(i,j)in{x_(1),dots,x_(n),y_(1),dots,y_(n)}v_{i, j} \in\left\{x_{1}, \ldots, x_{n}, y_{1}, \ldots, y_{n}\right\}vi,j∈{x1,…,xn,y1,…,yn} for every i , j i , j i,ji, ji,j (see [ 16 , 23 , 24 , 37 ] [ 16 , 23 , 24 , 37 ] [16,23,24,37][16,23,24,37][16,23,24,37] ). Unlike the CSP CSP CSP\operatorname{CSP}CSP, the problem QCSP ( Γ ) QCSP ⁡ ( Γ ) QCSP(Gamma)\operatorname{QCSP}(\Gamma)QCSP⁡(Γ) can be PSpace-hard if the constraint language Γ Î“ Gamma\GammaΓ is powerful enough. For example, QCSP ( { N A E } ) QCSP ⁡ ( { N A E } ) QCSP({NAE})\operatorname{QCSP}(\{\mathrm{NAE}\})QCSP⁡({NAE}) and Q C S P ( { 1 I N 3 } ) Q C S P ( { 1 I N 3 } ) QCSP({1IN3})\mathrm{QCSP}(\{1 \mathrm{IN} 3\})QCSP({1IN3}) on the domain A = { 0 , 1 } A = { 0 , 1 } A={0,1}A=\{0,1\}A={0,1} are PSpace-hard [25,27], and QCSP ( { } ) QCSP ⁡ ( { ≠ } ) QCSP({!=})\operatorname{QCSP}(\{\neq\})QCSP⁡({≠}) for | A | > 2 | A | > 2 |A| > 2|A|>2|A|>2 is also PSpace-hard [16]. Nevertheless, if Γ Î“ Gamma\GammaΓ consists of linear equations modulo p p ppp then QCSP ( Γ ) QCSP ⁡ ( Γ ) QCSP(Gamma)\operatorname{QCSP}(\Gamma)QCSP⁡(Γ) is tractable [16]. It was conjectured by Hubie Chen [ 2 2 , 2 4 ] [ 2 2 , 2 4 ] [22,24][\mathbf{2 2 , 2 4 ]}[22,24] that for any constraint language Γ Î“ Gamma\GammaΓ the problem QCSP ( Γ ) ( Γ ) (Gamma)(\Gamma)(Γ) is either solvable in polynomial time, or NP-complete, or PSpace-complete. Recently, this conjecture was disproved in [48], where the authors found constraint languages Γ Î“ Gamma\GammaΓ such that QCSP ( Γ ) QCSP ⁡ ( Γ ) QCSP(Gamma)\operatorname{QCSP}(\Gamma)QCSP⁡(Γ) is coNPcomplete (on a 3-element domain), DP-complete (on a 4-element domain), Θ 2 P Θ 2 P Theta_(2)^(P)\Theta_{2}^{P}Θ2P-complete (on a 10-element domain). Despite the whole zoo of the complexity classes, we still hope to obtain a full classification of the complexity for each constraint language Γ Î“ Gamma\GammaΓ.
Below we consider the main idea that makes the problem easier than PSpace.

3.1. PGP reduction for Π 2 Π 2 Pi_(2)\Pi_{2}Π2 restrictions

For simplicity let us consider the Π 2 Π 2 Pi_(2)\Pi_{2}Π2-restriction of QCSP ( Γ ) QCSP ⁡ ( Γ ) QCSP(Gamma)\operatorname{QCSP}(\Gamma)QCSP⁡(Γ), denoted QCSP 2 ( Γ ) QCSP 2 ⁡ ( Γ ) QCSP^(2)(Gamma)\operatorname{QCSP}^{2}(\Gamma)QCSP2⁡(Γ), in which the input is of the form
(3.1) x 1 x n y 1 y m R 1 ( ) R s ( ) (3.1) ∀ x 1 … ∀ x n ∃ y 1 … ∃ y m R 1 ( … ) ∧ ⋯ ∧ R s ( … ) {:(3.1)AAx_(1)dots AAx_(n)EEy_(1)dots EEy_(m)R_(1)(dots)^^cdots^^R_(s)(dots):}\begin{equation*} \forall x_{1} \ldots \forall x_{n} \exists y_{1} \ldots \exists y_{m} R_{1}(\ldots) \wedge \cdots \wedge R_{s}(\ldots) \tag{3.1} \end{equation*}(3.1)∀x1…∀xn∃y1…∃ymR1(…)∧⋯∧Rs(…)
Such an instance holds whenever the conjunctive formula R 1 ( ) R S ( ) R 1 ( … ) ∧ ⋯ ∧ R S ( … ) R_(1)(dots)^^cdots^^R_(S)(dots)R_{1}(\ldots) \wedge \cdots \wedge R_{S}(\ldots)R1(…)∧⋯∧RS(…) is solvable for any evaluation of x 1 , , x n x 1 , … , x n x_(1),dots,x_(n)x_{1}, \ldots, x_{n}x1,…,xn, which gives us a reduction of the instance to | A | n | A | n |A|^(n)|A|^{n}|A|n instances of CSP ( Γ ) CSP ⁡ Γ ∗ CSP(Gamma^(**))\operatorname{CSP}\left(\Gamma^{*}\right)CSP⁡(Γ∗), where by Γ Î“ ∗ Gamma^(**)\Gamma^{*}Γ∗ we denote Γ { ( x = a ) a A } Γ ∪ { ( x = a ) ∣ a ∈ A } Gamma uu{(x=a)∣a in A}\Gamma \cup\{(x=a) \mid a \in A\}Γ∪{(x=a)∣a∈A}. If we need to check | A | n | A | n |A|^(n)|A|^{n}|A|n tuples, which is exponentially many, this does not make the problem easier. Nevertheless, sometimes it is sufficient to check only polynomially many tuples. Let us consider a concrete example.
System of linear equations. Suppose A = { 0 , 1 } A = { 0 , 1 } A={0,1}A=\{0,1\}A={0,1} and Γ Î“ Gamma\GammaΓ consists of linear equations in Z 2 Z 2 Z_(2)\mathbb{Z}_{2}Z2. Let us check that the instance (3.1) holds for ( x 1 , , x n ) = ( 0 , , 0 ) x 1 , … , x n = ( 0 , … , 0 ) (x_(1),dots,x_(n))=(0,dots,0)\left(x_{1}, \ldots, x_{n}\right)=(0, \ldots, 0)(x1,…,xn)=(0,…,0), and ( x 1 , , x n ) = x 1 , … , x n = (x_(1),dots,x_(n))=\left(x_{1}, \ldots, x_{n}\right)=(x1,…,xn)= ( 0 , , 0 , 1 , 0 , , 0 ) ( 0 , … , 0 , 1 , 0 , … , 0 ) (0,dots,0,1,0,dots,0)(0, \ldots, 0,1,0, \ldots, 0)(0,…,0,1,0,…,0) for any position of 1 . To do this, we solve the CSP instance R 1 ( ) R 1 ( … ) ∧ R_(1)(dots)^^R_{1}(\ldots) \wedgeR1(…)∧ R s ( ) i = 1 n ( x i = 0 ) ⋯ ∧ R s ( … ) ∧ â‹€ i = 1 n   x i = 0 cdots^^R_(s)(dots)^^^^^_(i=1)^(n)(x_(i)=0)\cdots \wedge R_{s}(\ldots) \wedge \bigwedge_{i=1}^{n}\left(x_{i}=0\right)⋯∧Rs(…)∧⋀i=1n(xi=0), and for every j { 1 , 2 , , n } j ∈ { 1 , 2 , … , n } j in{1,2,dots,n}j \in\{1,2, \ldots, n\}j∈{1,2,…,n} we solve the instance R 1 ( ) R s ( ) ( x j = 1 ) i j ( x i = 0 ) R 1 ( … ) ∧ ⋯ ∧ R s ( … ) ∧ x j = 1 ∧ â‹€ i ≠ j   x i = 0 R_(1)(dots)^^cdots^^R_(s)(dots)^^(x_(j)=1)^^^^^_(i!=j)(x_(i)=0)R_{1}(\ldots) \wedge \cdots \wedge R_{s}(\ldots) \wedge\left(x_{j}=1\right) \wedge \bigwedge_{i \neq j}\left(x_{i}=0\right)R1(…)∧⋯∧Rs(…)∧(xj=1)∧⋀i≠j(xi=0). Each instance is a system of linear equations and can be solved in polynomial time. If at least one of the instances does not have a solution, then the instance (3.1) does not hold. Assume that all of them are satisfiable, then consider the relation Δ Î” Delta\DeltaΔ defined by the following pp-formula over Γ Î“ Gamma\GammaΓ :
Δ ( x 1 , , x n ) = y 1 y m R 1 ( ) R s ( ) Δ x 1 , … , x n = ∃ y 1 … ∃ y m R 1 ( … ) ∧ ⋯ ∧ R s ( … ) Delta(x_(1),dots,x_(n))=EEy_(1)dots EEy_(m)R_(1)(dots)^^cdots^^R_(s)(dots)\Delta\left(x_{1}, \ldots, x_{n}\right)=\exists y_{1} \ldots \exists y_{m} R_{1}(\ldots) \wedge \cdots \wedge R_{s}(\ldots)Δ(x1,…,xn)=∃y1…∃ymR1(…)∧⋯∧Rs(…)
Since Γ Î“ Gamma\GammaΓ is preserved by x + y + z , Δ x + y + z , Δ x+y+z,Deltax+y+z, \Deltax+y+z,Δ is also preserved by x + y + z x + y + z x+y+zx+y+zx+y+z. Applying this operation to the tuples ( 0 , 0 , , 0 ) , ( 1 , 0 , , 0 ) , ( 0 , 1 , 0 , , 0 ) , , ( 0 , 0 , , 0 , 1 ) Δ ( 0 , 0 , … , 0 ) , ( 1 , 0 , … , 0 ) , ( 0 , 1 , 0 , … , 0 ) , … , ( 0 , 0 , … , 0 , 1 ) ∈ Δ (0,0,dots,0),(1,0,dots,0),(0,1,0,dots,0),dots,(0,0,dots,0,1)in Delta(0,0, \ldots, 0),(1,0, \ldots, 0),(0,1,0, \ldots, 0), \ldots,(0,0, \ldots, 0,1) \in \Delta(0,0,…,0),(1,0,…,0),(0,1,0,…,0),…,(0,0,…,0,1)∈Δ coordinatewise, we derive that Δ = { 0 , 1 } n Δ = { 0 , 1 } n Delta={0,1}^(n)\Delta=\{0,1\}^{n}Δ={0,1}n, that is, Δ Î” Delta\DeltaΔ contains all tuples and (3.1) holds. Thus, we showed that QCSP 2 ( Γ ) QCSP 2 ⁡ ( Γ ) QCSP^(2)(Gamma)\operatorname{QCSP}^{2}(\Gamma)QCSP2⁡(Γ) is solvable in polynomial time.
This idea can be generalized as follows. We say that a set of operations F F FFF (or an algebra ( A ; F ) ) ( A ; F ) ) (A;F))(A ; F))(A;F)) has the polynomially generated powers (PGP) property if there exists a polynomial p ( n ) p ( n ) p(n)p(n)p(n) such that A n A n A^(n)A^{n}An can be generated from p ( n ) p ( n ) p(n)p(n)p(n) tuples using operations of F F FFF. Another behavior that might arise is that there is an exponential function f f fff so that the smallest generating sets for A n A n A^(n)A^{n}An require size at least f ( n ) f ( n ) f(n)f(n)f(n). We describe this as the exponentially generated powers (EGP) property. As it was proved in [43] these are the only two situations we could have on a finite domain. Moreover, it was shown that the generating set in the PGP case can be chosen to be very simple and efficiently computable. As a generating set of polynomial size, we can take the set of all tuples with at most k k kkk switches, where a switch is a position in ( a 1 , , a n ) a 1 , … , a n (a_(1),dots,a_(n))\left(a_{1}, \ldots, a_{n}\right)(a1,…,an) such that a i a i + 1 a i ≠ a i + 1 a_(i)!=a_(i+1)a_{i} \neq a_{i+1}ai≠ai+1. This gives a polynomial reduction of QCSP 2 ( Γ ) QCSP 2 ⁡ ( Γ ) QCSP^(2)(Gamma)\operatorname{QCSP}^{2}(\Gamma)QCSP2⁡(Γ) to CSP ( Γ ) CSP ⁡ Γ ∗ CSP(Gamma^(**))\operatorname{CSP}\left(\Gamma^{*}\right)CSP⁡(Γ∗) if Pol ( Γ ) Pol ⁡ ( Γ ) Pol(Gamma)\operatorname{Pol}(\Gamma)Pol⁡(Γ) has the P G P P G P PGP\mathrm{PGP}PGP property.

3.2. A general PGP reduction

Let us show that the same idea can be applied to the general form of QCSP ( Γ ) QCSP ⁡ ( Γ ) QCSP(Gamma)\operatorname{QCSP}(\Gamma)QCSP⁡(Γ). First, we show how to move universal quantifiers left and convert an instance into the Π 2 Π 2 Pi_(2)\Pi_{2}Π2-form. Notice that the sentence y 1 y 2 y s x Φ âˆƒ y 1 ∃ y 2 … ∃ y s ∀ x Φ EEy_(1)EEy_(2)dots EEy_(s)AA x Phi\exists y_{1} \exists y_{2} \ldots \exists y_{s} \forall x \Phi∃y1∃y2…∃ys∀xΦ is equivalent to
x 1 x 2 x | A | y 1 y 2 y s Φ 1 Φ 2 Φ | A | ∀ x 1 ∀ x 2 … ∀ x | A | ∃ y 1 ∃ y 2 … ∃ y s Φ 1 ∧ Φ 2 ∧ ⋯ ∧ Φ | A | AAx^(1)AAx^(2)dots AAx^(|A|)EEy_(1)EEy_(2)dots EEy_(s)Phi_(1)^^Phi_(2)^^cdots^^Phi_(|A|)\forall x^{1} \forall x^{2} \ldots \forall x^{|A|} \exists y_{1} \exists y_{2} \ldots \exists y_{s} \Phi_{1} \wedge \Phi_{2} \wedge \cdots \wedge \Phi_{|A|}∀x1∀x2…∀x|A|∃y1∃y2…∃ysΦ1∧Φ2∧⋯∧Φ|A|
where each Φ i Φ i Phi_(i)\Phi_{i}Φi is obtained from Φ Î¦ Phi\PhiΦ by renaming x x xxx by x i x i x^(i)x^{i}xi. In this way we can convert any instance y 1 x 1 y t x t Φ âˆƒ y 1 ∀ x 1 … ∃ y t ∀ x t Φ EEy_(1)AAx_(1)dots EEy_(t)AAx_(t)Phi\exists y_{1} \forall x_{1} \ldots \exists y_{t} \forall x_{t} \Phi∃y1∀x1…∃yt∀xtΦ of QCSP ( Γ ) QCSP ⁡ ( Γ ) QCSP(Gamma)\operatorname{QCSP}(\Gamma)QCSP⁡(Γ) into the Π 2 Π 2 Pi_(2)\Pi_{2}Π2-restriction by moving all universal quantifiers left:
x 1 1 x 1 | A | x 2 1 x 2 | A | 2 x t 1 x t | A | t (3.2) y 1 y 2 1 y 2 | A | y t 1 y t | A | t 1 Φ 1 Φ 2 Φ q ∀ x 1 1 … ∀ x 1 | A | ∀ x 2 1 … ∀ x 2 | A | 2 … ∀ x t 1 … ∀ x t | A | t (3.2) ∃ y 1 ∃ y 2 1 … ∃ y 2 | A | … ∃ y t 1 … ∃ y t | A | t − 1 Φ 1 ∧ Φ 2 ∧ ⋯ ∧ Φ q {:[AAx_(1)^(1)dots AAx_(1)^(|A|)AAx_(2)^(1)dots AAx_(2)^(|A|^(2))dots AAx_(t)^(1)dots AAx_(t)^(|A|^(t))],[(3.2)EEy_(1)EEy_(2)^(1)dots EEy_(2)^(|A|)dots EEy_(t)^(1)dots EEy_(t)^(|A|^(t-1))Phi_(1)^^Phi_(2)^^cdots^^Phi_(q)]:}\begin{align*} & \forall x_{1}^{1} \ldots \forall x_{1}^{|A|} \forall x_{2}^{1} \ldots \forall x_{2}^{|A|^{2}} \ldots \forall x_{t}^{1} \ldots \forall x_{t}^{|A|^{t}} \\ & \exists y_{1} \exists y_{2}^{1} \ldots \exists y_{2}^{|A|} \ldots \exists y_{t}^{1} \ldots \exists y_{t}^{|A|^{t-1}} \Phi_{1} \wedge \Phi_{2} \wedge \cdots \wedge \Phi_{q} \tag{3.2} \end{align*}∀x11…∀x1|A|∀x21…∀x2|A|2…∀xt1…∀xt|A|t(3.2)∃y1∃y21…∃y2|A|…∃yt1…∃yt|A|t−1Φ1∧Φ2∧⋯∧Φq
where each Φ i Φ i Phi_(i)\Phi_{i}Φi is obtained from Φ Î¦ Phi\PhiΦ by renaming the variables. The only problem with this reduction is that the number of variables and constraints could be exponential. Nevertheless, we can apply the PGP idea to this sentence. If Pol ( Γ ) Pol ⁡ ( Γ ) Pol(Gamma)\operatorname{Pol}(\Gamma)Pol⁡(Γ) has the PGP property then there exists a constant k k kkk such that it is sufficient to check (3.2) only on the tuples with at most k k kkk switches. Those k k kkk switches appear in at most k k kkk original x x xxx-variables and all the remaining variables can be fixed with constants. This allows reducing QCSP ( Γ ) QCSP ⁡ ( Γ ) QCSP(Gamma)\operatorname{QCSP}(\Gamma)QCSP⁡(Γ) to a sentence with a constant number of universal quantifiers or even remove all of them.
Theorem 3.1 ([45]). Suppose Pol ( Γ ) Pol ⁡ ( Γ ) Pol(Gamma)\operatorname{Pol}(\Gamma)Pol⁡(Γ) has the PGP property. Then QCSP ( Γ ) QCSP ⁡ ( Γ ) QCSP(Gamma)\operatorname{QCSP}(\Gamma)QCSP⁡(Γ) is polynomially equivalent to the modification of QCSP 2 ( Γ ) QCSP 2 ⁡ ( Γ ) QCSP^(2)(Gamma)\operatorname{QCSP}^{2}(\Gamma)QCSP2⁡(Γ) where sentences have at most | A | | A | |A||A||A| universally quantified variables.
Theorem 3.2 ([45]). Suppose Pol ( Γ ) Pol ⁡ ( Γ ) Pol(Gamma)\operatorname{Pol}(\Gamma)Pol⁡(Γ) has the P G P P G P PGPP G PPGP property. Then QCSP ( Γ ) QCSP ⁡ ( Γ ) QCSP(Gamma)\operatorname{QCSP}(\Gamma)QCSP⁡(Γ) is polynomially reduced to CSP ( Γ ) CSP ⁡ Γ ∗ CSP(Gamma^(**))\operatorname{CSP}\left(\Gamma^{*}\right)CSP⁡(Γ∗).
This idea gives us a complete classification of the complexity of QCSP ( Γ ) QCSP ⁡ ( Γ ) QCSP(Gamma)\operatorname{QCSP}(\Gamma)QCSP⁡(Γ) for a two-element domain.
Theorem 3.3 ([25, 27]). Suppose Γ Î“ Gamma\GammaΓ is a constraint language on { 0 , 1 } { 0 , 1 } {0,1}\{0,1\}{0,1}. Then QCSP ( Γ ) QCSP ⁡ ( Γ ) QCSP(Gamma)\operatorname{QCSP}(\Gamma)QCSP⁡(Γ) is solvable in polynomial time if Γ Î“ Gamma\GammaΓ is preserved by an idempotent WNU; QCSP ( Γ ) QCSP ⁡ ( Γ ) QCSP(Gamma)\operatorname{QCSP}(\Gamma)QCSP⁡(Γ) is PSpacecomplete otherwise.
It is known [39] that if Γ Î“ Gamma\GammaΓ admits an idempotent WNU, then it is preserved by x + y + z , x y , x y x + y + z , x ∨ y , x ∧ y x+y+z,x vv y,x^^yx+y+z, x \vee y, x \wedge yx+y+z,x∨y,x∧y, or x y y z x z x y ∨ y z ∨ x z xy vv yz vv xzx y \vee y z \vee x zxy∨yz∨xz. Hence, to prove the above theorem, it is sufficient to check that these operations guarantee the PGP property, which by Theorem 3.2 gives a polynomial reduction to a tractable CSP. To show the PGP property, we verify that the tuples ( 0 , 0 , , 0 ) , ( 1 , 1 , , 1 ) , ( 1 , 0 , , 0 ) , ( 0 , 1 , 0 , , 0 ) , , ( 0 , , 0 , 1 ) ( 0 , 0 , … , 0 ) , ( 1 , 1 , … , 1 ) , ( 1 , 0 , … , 0 ) , ( 0 , 1 , 0 , … , 0 ) , … , ( 0 , … , 0 , 1 ) (0,0,dots,0),(1,1,dots,1),(1,0,dots,0),(0,1,0,dots,0),dots,(0,dots,0,1)(0,0, \ldots, 0),(1,1, \ldots, 1),(1,0, \ldots, 0),(0,1,0, \ldots, 0), \ldots,(0, \ldots, 0,1)(0,0,…,0),(1,1,…,1),(1,0,…,0),(0,1,0,…,0),…,(0,…,0,1) generate { 0 , 1 } n { 0 , 1 } n {0,1}^(n)\{0,1\}^{n}{0,1}n using any of the above operations.

3.3. Does EGP mean hard?

Thus, if Pol ( Γ ) Pol ⁡ ( Γ ) Pol(Gamma)\operatorname{Pol}(\Gamma)Pol⁡(Γ) has the PGP property then we have a nice reduction to CSP, and QCSP ( Γ ) QCSP ⁡ ( Γ ) QCSP(Gamma)\operatorname{QCSP}(\Gamma)QCSP⁡(Γ) belongs to NP. What can we say about the complexity of QCSP ( Γ ) QCSP ⁡ ( Γ ) QCSP(Gamma)\operatorname{QCSP}(\Gamma)QCSP⁡(Γ) if Pol ( Γ ) Pol ⁡ ( Γ ) Pol(Gamma)\operatorname{Pol}(\Gamma)Pol⁡(Γ) has the EGP property? Hubie Chen conjectured in [24] that Q C S P ( Γ ) Q C S P ( Γ ) QCSP(Gamma)\mathrm{QCSP}(\Gamma)QCSP(Γ) is PSpace-complete whenever Pol ( Γ ) Pol ⁡ ( Γ ) Pol(Gamma)\operatorname{Pol}(\Gamma)Pol⁡(Γ) has the EGP property.
For constraint languages Γ Î“ Gamma\GammaΓ containing all constant relations, a characterization of Pol ( Γ ) Pol ⁡ ( Γ ) Pol(Gamma)\operatorname{Pol}(\Gamma)Pol⁡(Γ) that have the EGP property is given in [43], where it is shown that Γ Î“ Gamma\GammaΓ must allow the pp-definition of relations τ n Ï„ n tau_(n)\tau_{n}Ï„n with the following special form.
Definition 3.4. Let α β = A α ∪ β = A alpha uu beta=A\alpha \cup \beta=Aα∪β=A, yet neither α α alpha\alphaα nor β β beta\betaβ equals D D DDD. Let S = α 3 β 3 S = α 3 ∪ β 3 S=alpha^(3)uubeta^(3)S=\alpha^{3} \cup \beta^{3}S=α3∪β3 and τ n Ï„ n tau_(n)\tau_{n}Ï„n be the 3 n 3 n 3n3 n3n-ary relation given by i = 1 n S ( x i , y i , z i ) ⋁ i = 1 n   S x i , y i , z i vvv_(i=1)^(n)S(x_(i),y_(i),z_(i))\bigvee_{i=1}^{n} S\left(x_{i}, y_{i}, z_{i}\right)⋁i=1nS(xi,yi,zi).
The complement to S S SSS represents the not-all-equal relation and the relations τ n Ï„ n tau_(n)\tau_{n}Ï„n allow for the encoding of the complement of Not-All-Equal 3-Satisfiability (where α β α ∖ β alpha\\beta\alpha \backslash \betaα∖β is 0 and β α β ∖ α beta\\alpha\beta \backslash \alphaβ∖α is 1). Thus, if one has polynomially computable (in n n nnn ) pp-definitions of τ n Ï„ n tau_(n)\tau_{n}Ï„n, then it is clear that QCSP ( Γ ) QCSP ⁡ ( Γ ) QCSP(Gamma)\operatorname{QCSP}(\Gamma)QCSP⁡(Γ) is co-NP-hard [22]. In light of this observation, it seemed that only a small step remained to proving the actual Chen Conjecture, at least with coNP-hard in place of PSpace-complete.

3.4. Surprising constraint language and the QCSP on a 3-element domain

As we saw in Section 2.7, the CSP is NP-hard if and only the we can pp-define (pp-construct) the not-all-equal relation. In the previous subsection, we mentioned that in the EGP case we can always pp-construct the complement to Not-All-Equal 3-Satisfability, which almost guarantees coNP-hardness. Surprisingly, two constraint languages Γ Î“ Gamma\GammaΓ on A = { 0 , 1 , 2 } A = { 0 , 1 , 2 } A={0,1,2}A=\{0,1,2\}A={0,1,2} were discovered in [48] for which any pp-definition of τ n Ï„ n tau_(n)\tau_{n}Ï„n is of exponential size, which makes it impossible to use this reduction.
Theorem 3.5 ([48]). There exists a constraint language Γ Î“ Gamma\GammaΓ on { 0 , 1 , 2 } { 0 , 1 , 2 } {0,1,2}\{0,1,2\}{0,1,2} such that
(1) Pol ( Γ ) Pol ⁡ ( Γ ) Pol(Gamma)\operatorname{Pol}(\Gamma)Pol⁡(Γ) has the EGP property,
(2) τ n Ï„ n tau_(n)\tau_{n}Ï„n is pp-definable over Γ Î“ Gamma\GammaΓ
(3) any pp-definition of τ n Ï„ n tau_(n)\tau_{n}Ï„n for α = { 0 , 1 } α = { 0 , 1 } alpha={0,1}\alpha=\{0,1\}α={0,1} and β = { 0 , 2 } β = { 0 , 2 } beta={0,2}\beta=\{0,2\}β={0,2} has at least 2 n 2 n 2^(n)2^{n}2n variables, and
(4) QCSP ( Γ ) QCSP ⁡ ( Γ ) QCSP(Gamma)\operatorname{QCSP}(\Gamma)QCSP⁡(Γ) is solvable in polynomial time.
The algorithm in (4) consists of the following three steps. First, it reduces an instance to a Π 2 Π 2 Pi_(2)\Pi_{2}Π2-form x 1 x n y 1 y m Φ âˆ€ x 1 … ∀ x n ∃ y 1 … ∃ y m Φ AAx_(1)dots AAx_(n)EEy_(1)dots EEy_(m)Phi\forall x_{1} \ldots \forall x_{n} \exists y_{1} \ldots \exists y_{m} \Phi∀x1…∀xn∃y1…∃ymΦ. Then, by solving polynomially many CSPs, it calculates polynomially many evaluations to ( x 1 , , x n ) x 1 , … , x n (x_(1),dots,x_(n))\left(x_{1}, \ldots, x_{n}\right)(x1,…,xn) we need to check. Finally, it checks that Φ Î¦ Phi\PhiΦ has a solution for each of these evaluations. It is proved in [48] that this test guarantees that the instance holds.
This result was shocking because of several reasons. Not only it disproved the widely believed Chen Conjecture but showed that we need to worry about the existence of an efficient pp-definition. Before, if we could pp-define a strong relation (such as τ n Ï„ n tau_(n)\tau_{n}Ï„n ) then the problem was hard. Another surprising thing is that we have to calculate the evaluations of ( x 1 , , x n ) x 1 , … , x n (x_(1),dots,x_(n))\left(x_{1}, \ldots, x_{n}\right)(x1,…,xn) we need to check, In fact, if we do not look inside Φ Î¦ Phi\PhiΦ then we have to check all the tuples from { 0 , 1 } n { 0 , 1 } n {0,1}^(n)\{0,1\}^{n}{0,1}n.
Despite the fact that we are far from having a full classification of the complexity of the QCSP, we know the complexity for any constraint language on a 3-element domain containing all constant relations. This classification is given in terms of polymorphisms.
Theorem 3.6 ([48]). Suppose Γ Î“ Gamma\GammaΓ is a finite constraint language on { 0 , 1 , 2 } { 0 , 1 , 2 } {0,1,2}\{0,1,2\}{0,1,2} containing all constant relations. Then Q C S P ( Γ ) Q C S P ( Γ ) QCSP(Gamma)\mathrm{QCSP}(\Gamma)QCSP(Γ) is either solvable in polynomial time, N P N P NPN PNP-complete, coNPcomplete, or PSpace-complete.

3.5. Conclusions

Unlike the CSP where the complexity is known for any constraint language Γ Î“ Gamma\GammaΓ here the complexity is wide open.
Problem 2. What is the complexity of QCSP ( Γ ) QCSP ⁡ ( Γ ) QCSP(Gamma)\operatorname{QCSP}(\Gamma)QCSP⁡(Γ) ?
Moreover, we do not even have a conjecture describing the complexity. We know that for some constraint languages Γ Î“ Gamma\GammaΓ the problem QCSP ( Γ ) QCSP ⁡ ( Γ ) QCSP(Gamma)\operatorname{QCSP}(\Gamma)QCSP⁡(Γ) is DP-complete and Θ 2 P Θ 2 P Theta_(2)^(P)\Theta_{2}^{P}Θ2P-complete, but we do not know whether there are some other complexity classes and whether we have finitely many of them.
Problem 3. What complexity classes (up to polynomial equivalence) can be expressed as QCSP ( Γ ) QCSP ⁡ ( Γ ) QCSP(Gamma)\operatorname{QCSP}(\Gamma)QCSP⁡(Γ) for some constraint language Γ Î“ Gamma\GammaΓ ?
Now it is hard to believe that there will be a simple classification, that is why it is interesting to start with a 3-element domain (without constant relations) and 4-element domain. Probably, a more important problem is to describe all tractable cases assuming P N P P ≠ N P P!=NP\mathrm{P} \neq \mathrm{NP}P≠NP.
Problem 4. Describe all constraint languages Γ Î“ Gamma\GammaΓ such that QCSP ( Γ ) QCSP ⁡ ( Γ ) QCSP(Gamma)\operatorname{QCSP}(\Gamma)QCSP⁡(Γ) is solvable in polynomial time.

4. OTHER VARIANTS OF CSP

The Quantified CSP is only one of many other variations and generalizations of the CSP whose complexity is still unknown. Here we list some of them.

4.1. CSP over an infinite domain

If we consider CSP ( Γ ) CSP ⁡ ( Γ ) CSP(Gamma)\operatorname{CSP}(\Gamma)CSP⁡(Γ) for a constraint language on an infinite domain, the situation changes significantly. As was shown in [11], every computational problem is equivalent (under polynomial-time Turing reductions) to a problem of the form CSP ( Γ ) CSP ⁡ ( Γ ) CSP(Gamma)\operatorname{CSP}(\Gamma)CSP⁡(Γ). In [14] the authors gave a nice example of a constraint language Γ Î“ Gamma\GammaΓ such that CSP ( Γ ) CSP ⁡ ( Γ ) CSP(Gamma)\operatorname{CSP}(\Gamma)CSP⁡(Γ) is undecidable. Let Γ Î“ Gamma\GammaΓ consist of three relations (predicates) x + y = z , x y = z x + y = z , x â‹… y = z x+y=z,x*y=zx+y=z, x \cdot y=zx+y=z,xâ‹…y=z and x = 1 x = 1 x=1x=1x=1 over the set of all integers Z Z Z\mathbb{Z}Z. Then the Hilbert's 10th problem can be expressed as CSP ( Γ ) CSP ⁡ ( Γ ) CSP(Gamma)\operatorname{CSP}(\Gamma)CSP⁡(Γ), which proves undecidability of CSP ( Γ ) CSP ⁡ ( Γ ) CSP(Gamma)\operatorname{CSP}(\Gamma)CSP⁡(Γ). Nevertheless, there are additional assumptions that send the CSP back to the class NP and make complexity classifications possible [8,12]. For more information about the infinite domain CSP and the algebraic approach, see [10,14].

4.2. Surjective Constraint Satisfaction Problem

A natural modification of the CSP is the Surjective Constraint Satisfaction Problem, where we want to find a surjective solution. Formally, for a constraint language Γ Î“ Gamma\GammaΓ over a domain A , SCSP ( Γ ) A , SCSP ⁡ ( Γ ) A,SCSP(Gamma)A, \operatorname{SCSP}(\Gamma)A,SCSP⁡(Γ) is the following decision problem: given a formula
R 1 ( ) R s ( ) R 1 ( … ) ∧ ⋯ ∧ R s ( … ) R_(1)(dots)^^cdots^^R_(s)(dots)R_{1}(\ldots) \wedge \cdots \wedge R_{s}(\ldots)R1(…)∧⋯∧Rs(…)
where all relations R 1 , , R s R 1 , … , R s R_(1),dots,R_(s)R_{1}, \ldots, R_{s}R1,…,Rs are from Γ Î“ Gamma\GammaΓ, decide whether there exists a surjective solution, that is, a solution with { x 1 , , x n } = A x 1 , … , x n = A {x_(1),dots,x_(n)}=A\left\{x_{1}, \ldots, x_{n}\right\}=A{x1,…,xn}=A. Probably, the most natural examples of the Surjective CSP are defined as the surjective graph homomorphism problem, which is equivalent to SCSP ( Γ ) SCSP ⁡ ( Γ ) SCSP(Gamma)\operatorname{SCSP}(\Gamma)SCSP⁡(Γ) where Γ Î“ Gamma\GammaΓ consists of one binary relation that is viewed as a graph. An interesting fact about the complexity of the Surjective CSP is that its complexity remained unknown for many years even for very simple graphs and constraint languages. Three most popular examples of such long-standing problems are the complexity for the reflexive 4-cycle (undirected having a loop at each vertex) [38], the complexity for the nonreflexive 6-cycle (undirected without loops) [41], and the complexity of the No-Rainbow-Problem ( SCSP ( { N } ) ( SCSP ⁡ ( { N } ) (SCSP({N})(\operatorname{SCSP}(\{N\})(SCSP⁡({N}) where A = { 0 , 1 , 2 } A = { 0 , 1 , 2 } A={0,1,2}A=\{0,1,2\}A={0,1,2} and N = { ( a , b , c ) { a , b , c } A } ) N = { ( a , b , c ) ∣ { a , b , c } ≠ A } ) N={(a,b,c)∣{a,b,c}!=A})N=\{(a, b, c) \mid\{a, b, c\} \neq A\})N={(a,b,c)∣{a,b,c}≠A}) [46]. Even though these three problems turned out to be NP-complete, the complexity seems to be unknown even for graphs of size 5 and cycles.
Problem 5. What is the complexity of SCSP ( Γ ) SCSP ⁡ ( Γ ) SCSP(Gamma)\operatorname{SCSP}(\Gamma)SCSP⁡(Γ) ?
It was shown in [46] that the complexity of SCSP ( Γ ) SCSP ⁡ ( Γ ) SCSP(Gamma)\operatorname{SCSP}(\Gamma)SCSP⁡(Γ) cannot be described in terms of polymorphisms, which disproved the only conjecture about the complexity of SCSP ( Γ ) SCSP ⁡ ( Γ ) SCSP(Gamma)\operatorname{SCSP}(\Gamma)SCSP⁡(Γ) we know. This conjecture, formulated by Hubie Chen, says that SCSP ( Γ ) SCSP ⁡ ( Γ ) SCSP(Gamma)\operatorname{SCSP}(\Gamma)SCSP⁡(Γ) and CSP ( Γ ) CSP ⁡ Γ ∗ CSP(Gamma^(**))\operatorname{CSP}\left(\Gamma^{*}\right)CSP⁡(Γ∗) have the same complexity. Nevertheless, this conjecture still can hold for graphs.
Problem 6. Is it true that SCSP ( { R } ) SCSP ⁡ ( { R } ) SCSP({R})\operatorname{SCSP}(\{R\})SCSP⁡({R}) and CSP ( { R } ) CSP ⁡ { R } ∗ CSP({R}^(**))\operatorname{CSP}\left(\{R\}^{*}\right)CSP⁡({R}∗) have the same complexity for any binary relation R R RRR ?
For more results on the complexity of the SCSP, see the survey [13].

4.3. Promise CSP

A natural generalization of the CSP is the Promise Constraint Satisfaction Problem, where a promise about the input is given (see [18,21]). Let Γ = { ( R 1 A , R 1 B ) , , ( R t A , R t B ) } Γ = R 1 A , R 1 B , … , R t A , R t B Gamma={(R_(1)^(A),R_(1)^(B)),dots,(R_(t)^(A),R_(t)^(B))}\Gamma=\left\{\left(R_{1}^{A}, R_{1}^{B}\right), \ldots,\left(R_{t}^{A}, R_{t}^{B}\right)\right\}Γ={(R1A,R1B),…,(RtA,RtB)}, where R i A R i A R_(i)^(A)R_{i}^{A}RiA and R i B R i B R_(i)^(B)R_{i}^{B}RiB are relations of the same arity over the domains A A AAA and B B BBB, respectively. Then PCSP ( Γ ) PCSP ⁡ ( Γ ) PCSP(Gamma)\operatorname{PCSP}(\Gamma)PCSP⁡(Γ) is the following decision problem: given two conjunctive formulas
R i 1 A ( v 1 , 1 , , v 1 , n 1 ) R i s A ( v s , 1 , , v s , n s ) R i 1 B ( v 1 , 1 , , v 1 , n 1 ) R i s B ( v s , 1 , , v s , n s ) R i 1 A v 1 , 1 , … , v 1 , n 1 ∧ ⋯ ∧ R i s A v s , 1 , … , v s , n s R i 1 B v 1 , 1 , … , v 1 , n 1 ∧ ⋯ ∧ R i s B v s , 1 , … , v s , n s {:[R_(i_(1))^(A)(v_(1,1),dots,v_(1,n_(1)))^^cdots^^R_(i_(s))^(A)(v_(s,1),dots,v_(s,n_(s)))],[R_(i_(1))^(B)(v_(1,1),dots,v_(1,n_(1)))^^cdots^^R_(i_(s))^(B)(v_(s,1),dots,v_(s,n_(s)))]:}\begin{aligned} & R_{i_{1}}^{A}\left(v_{1,1}, \ldots, v_{1, n_{1}}\right) \wedge \cdots \wedge R_{i_{s}}^{A}\left(v_{s, 1}, \ldots, v_{s, n_{s}}\right) \\ & R_{i_{1}}^{B}\left(v_{1,1}, \ldots, v_{1, n_{1}}\right) \wedge \cdots \wedge R_{i_{s}}^{B}\left(v_{s, 1}, \ldots, v_{s, n_{s}}\right) \end{aligned}Ri1A(v1,1,…,v1,n1)∧⋯∧RisA(vs,1,…,vs,ns)Ri1B(v1,1,…,v1,n1)∧⋯∧RisB(vs,1,…,vs,ns)
where ( R i j A , R i j B ) R i j A , R i j B (R_(i_(j))^(A),R_(i_(j))^(B))\left(R_{i_{j}}^{A}, R_{i_{j}}^{B}\right)(RijA,RijB) are from Γ Î“ Gamma\GammaΓ for every j j jjj and v i , j { x 1 , , x n } v i , j ∈ x 1 , … , x n v_(i,j)in{x_(1),dots,x_(n)}v_{i, j} \in\left\{x_{1}, \ldots, x_{n}\right\}vi,j∈{x1,…,xn} for every i , j i , j i,ji, ji,j, distinguish between the case when both of them are satisfiable, and when both of them are not satisfiable. Thus, we are given two CSP instances and a promise that if one has a solution then another has a solution. Usually, it is also assumed that there exists a mapping (homomorphism) h : A B h : A → B h:A rarr Bh: A \rightarrow Bh:A→B such that h ( R i A ) R i B h R i A ⊆ R i B h(R_(i)^(A))subeR_(i)^(B)h\left(R_{i}^{A}\right) \subseteq R_{i}^{B}h(RiA)⊆RiB for every i i iii. In this case, the satisfiability of the first formula implies
the satisfiability of the second one. To make sure that the promise can actually make an NP-hard problem tractable, see Example 2.8 in [21].
The most popular example of the Promise CSP is graph ( k , l ) ( k , l ) (k,l)(k, l)(k,l)-colorability, where we need to distinguish between k k kkk-colorable graphs and not even l l lll-colorable, where k l k ≤ l k <= lk \leq lk≤l. This problem can be written as follows.
Problem 7. Let | A | = k , | B | = l , Γ = { ( A , B ) } | A | = k , | B | = l , Γ = ≠ A , ≠ B |A|=k,|B|=l,Gamma={(!=_(A),!=_(B))}|A|=k,|B|=l, \Gamma=\left\{\left(\neq_{A}, \neq_{B}\right)\right\}|A|=k,|B|=l,Γ={(≠A,≠B)}. What is the complexity of PCSP ( Γ ) PCSP ⁡ ( Γ ) PCSP(Gamma)\operatorname{PCSP}(\Gamma)PCSP⁡(Γ) ?
Recently, it was proved [21] that ( k , l ) ( k , l ) (k,l)(k, l)(k,l)-colorability is NP-hard for l = 2 k 1 l = 2 k − 1 l=2k-1l=2 k-1l=2k−1 and k 3 k ≥ 3 k >= 3k \geq 3k≥3 but even the complexity of ( 3 , 6 ) ( 3 , 6 ) (3,6)(3,6)(3,6)-colorability is still not known.
Even for a 2-element domain the problem is wide open, but recently a dichotomy for symmetric Boolean PCSP was proved [30].
Problem 8. Let A = B = { 0 , 1 } A = B = { 0 , 1 } A=B={0,1}A=B=\{0,1\}A=B={0,1}. What is the complexity of PCSP ( Γ ) PCSP ⁡ ( Γ ) PCSP(Gamma)\operatorname{PCSP}(\Gamma)PCSP⁡(Γ) ?

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DMITRIY ZHUK

Department of Mechanics and Mathematics, Lomonosov Moscow State University, Vorobjovy Gory, 119899 Moscow, Russia, zhuk @intsys.msu.ru
  1. ALGEBRA

A TOTALLY DISCONNECTED INVITATION TO LOCALLY COMPACT GROUPS

PIERRE-EMMANUEL CAPRACE ANDGEORGE A. WILLIS

ABSTRACT

We present a selection of results contributing to a structure theory of totally disconnected locally compact groups.

MATHEMATICS SUBJECT CLASSIFICATION 2020

Primary 22D05; Secondary 20E15, 20E32, 20E08, 22F50, 43A07

KEYWORDS

Locally compact group, profinite group, amenable group, simple group, topological dynamics, commensurated subgroup, minimising subgroup, flat group of automorphisms

1. INTRODUCTION

Locally compact groups have attracted sustained attention because, on the one hand, rich classes of these groups have fruitful connections with other fields and, on the other, they have a well-developed theory that underpins those connections and delineates group structure. Salient features of this theory are the existence of a left-invariant, or Haar, measure; and the decomposition of a general group into pieces, many of which may be described concretely and in detail.
Haar measure permits representations of a general locally compact group by operators on spaces of measurable functions, and is thus the foundation for abstract harmonic analysis. Connections with partial differential equations, physics, and number theory come about through these representations. Locally compact groups are the largest class for which an invariant measure exists and for which harmonic analysis can be done in this form, as was shown by A. Weil [81].
The decomposition theory of an arbitrary locally compact group G G GGG begins with the short exact sequence
0 G G G / G 0 0 → G ∘ → G → G / G ∘ → 0 0rarrG^(@)rarr G rarr G//G^(@)rarr00 \rightarrow G^{\circ} \rightarrow G \rightarrow G / G^{\circ} \rightarrow 00→G∘→G→G/G∘→0
in which the closed normal subgroup G G ∘ G^(@)G^{\circ}G∘ is the connected component of the identity. The Gleason-Yamabe theorem [73, тн. 6.0.11] applies to G G ∘ G^(@)G^{\circ}G∘ to show that it is a projective limit of connected Lie groups, and powerful tools from the theory of Lie groups may thus be brought to bear on G G ∘ G^(@)G^{\circ}G∘. Groups occurring in physics and differential equations are often Lie groups. The quotient G / G G / G ∘ G//G^(@)G / G^{\circ}G/G∘ is a totally disconnected locally compact group (abbreviated tdlc group). Lie groups over local fields are important examples of tdlc groups having links to number theory and algebraic geometry (see, for example, [ 49 , 69 ] [ 49 , 69 ] [49,69][49,69][49,69] ). Unlike the connected case however, many other significant tdlc groups, such as the automorphism groups of locally finite trees first studied in [76], cannot be approximated by Lie groups. While substantial progress has been made with our understanding of tdlc groups much remains to be done before it could be said that the structure theory has reached maturity. This article surveys our current state of knowledge, much of which is founded on a theorem of van Dantzig, [77], which ensures that a tdlc group G G GGG has a basis of identity neighborhoods consisting of compact open subgroups.
Decompositions of general tdlc groups are described in Section 2. This section includes a discussion of the so-called elementary groups, which are those built from discrete and compact groups by standard operations. Discrete and compact groups are large domains of study in their own right and it is seen how elementary groups can be factored out in the analysis of a general tdlc group. Simple groups are an important aspect of any decomposition theory and what is known about them is summarized in Section 3. This includes a local structure theory and the extent to which local structure determines the global structure of the group. Section 4 treats scale methods, which associate invariants and special subgroups to abelian groups of automorphisms and which in some circumstances substitute for the Lie methods available for connected groups. A unifying theme of our approach is the dynamics of the conjugation action: Section 2 is concerned with the conjugation action of G G GGG on its closed subnormal subgroups, Section 3 uses in an essential way the conjugation
action of G G GGG on its closed subgroups, especially those that are locally normal, while Section 4 concerns the dynamics of the conjugation action of cyclic subgroups (and, more generally, flat subgroups) on the topological space G G GGG. Section 5 highlights a few open questions and directions for further research.

2. DECOMPOSITION THEORY

2.1. Normal subgroup structure

Finite groups, Lie groups, and algebraic groups constitute three of the most important classes of groups. Their respective structure theories are deep and far-reaching. One of the common themes consists in reducing problems concerning a given group G G GGG in one of these classes to problems about simple groups in the corresponding class, and then tackling the reduced problem by invoking classification results. Striking illustrations of this approach in the case of finite groups can be consulted in R. Guralnick's ICM address [43].
Since the category of locally compact groups contains all discrete groups, hence all groups, developing a similar theory for locally compact groups is hopeless. Nevertheless, the possibility to construct meaningful "decompositions of locally compact groups into simple pieces" has been highlighted in [23]. Wide-ranging results have subsequently been established by C. Reid and P. Wesolek in a series of papers [63,64], some of whose contributions are summarized below. A more in-depth survey can be consulted in [62].
Given closed normal subgroups K , L K , L K,LK, LK,L of a locally compact group G G GGG, the quotient group K / L K / L K//LK / LK/L is called a chief factor of G G GGG if L L LLL is strictly contained in K K KKK and for every closed normal subgroup N N NNN of G G GGG with L N K L ≤ N ≤ K L <= N <= KL \leq N \leq KL≤N≤K, we have N = L N = L N=LN=LN=L or N = K N = K N=KN=KN=K. Given a closed normal subgroup N N NNN of G G GGG, the quotient Q = G / N Q = G / N Q=G//NQ=G / NQ=G/N is a chief factor if and only if Q Q QQQ is topologically simple, i.e., Q Q QQQ is nontrivial and the only closed normal subgroups of Q Q QQQ are { 1 } { 1 } {1}\{1\}{1} and Q Q QQQ. More generally, every chief factor Q = K / L Q = K / L Q=K//LQ=K / LQ=K/L is topologically characteristically simple, i.e., the only closed subgroups of Q Q QQQ that are invariant under all homeomorphic automorphisms of Q Q QQQ are { 1 } { 1 } {1}\{1\}{1} and Q Q QQQ. A topological group is called compactly generated if it has a compact generating set.
Theorem 2.1 (See [64, тн. 1.3]). Every compactly generated tdlc group G G GGG has a finite series { 1 } = G 0 < G 1 < G 2 < < G n = G { 1 } = G 0 < G 1 < G 2 < ⋯ < G n = G {1}=G_(0) < G_(1) < G_(2) < cdots < G_(n)=G\{1\}=G_{0}<G_{1}<G_{2}<\cdots<G_{n}=G{1}=G0<G1<G2<⋯<Gn=G of closed normal subgroups such that for all i = 1 , , n i = 1 , … , n i=1,dots,ni=1, \ldots, ni=1,…,n, the quotient G i / G i 1 G i / G i − 1 G_(i)//G_(i-1)G_{i} / G_{i-1}Gi/Gi−1 is compact, or discrete infinite, or a chief factor of G G GGG which is noncompact, nondiscrete, and second countable.
A normal series as in Theorem 2.1 is called an essentially chief series. The theorem obviously has no content if G G GGG is compact or discrete. Let us illustrate Theorem 2.1 with two examples.
Example 2.2. Let I I III be a set and for each i I i ∈ I i in Ii \in Ii∈I, let G i G i G_(i)G_{i}Gi be a tdlc group and U i G i U i ≤ G i U_(i) <= G_(i)U_{i} \leq G_{i}Ui≤Gi be a compact open subgroup. The restricted product of ( G i , U i ) i I G i , U i i ∈ I (G_(i),U_(i))_(i in I)\left(G_{i}, U_{i}\right)_{i \in I}(Gi,Ui)i∈I, denoted by i I ( G i , U i ) ⨁ i ∈ I   G i , U i bigoplus_(i in I)(G_(i),U_(i))\bigoplus_{i \in I}\left(G_{i}, U_{i}\right)⨁i∈I(Gi,Ui), is the subgroup of i I G i ∏ i ∈ I   G i prod_(i in I)G_(i)\prod_{i \in I} G_{i}∏i∈IGi consisting of those tuples ( g i ) i I g i i ∈ I (g_(i))_(i in I)\left(g_{i}\right)_{i \in I}(gi)i∈I such that g i U i g i ∈ U i g_(i)inU_(i)g_{i} \in U_{i}gi∈Ui for all but finitely many i I i ∈ I i in Ii \in Ii∈I. It is endowed with the unique tdlc group topology such that
the inclusion i I U i i I ( G i , U i ) ∏ i ∈ I   U i → ⨁ i ∈ I   G i , U i prod_(i in I)U_(i)rarrbigoplus_(i in I)(G_(i),U_(i))\prod_{i \in I} U_{i} \rightarrow \bigoplus_{i \in I}\left(G_{i}, U_{i}\right)∏i∈IUi→⨁i∈I(Gi,Ui) is continuous and open. Given a prime p p ppp, set M ( p ) = n Z ( P S L 2 ( Q p ) , P S L 2 ( Z p ) ) M ( p ) = ⨁ n ∈ Z   P S L 2 Q p , P S L 2 Z p M(p)=bigoplus_(n inZ)(PSL_(2)(Q_(p)),PSL_(2)(Z_(p)))M(p)=\bigoplus_{n \in \mathbf{Z}}\left(\mathrm{PSL}_{2}\left(\mathbf{Q}_{p}\right), \mathrm{PSL}_{2}\left(\mathbf{Z}_{p}\right)\right)M(p)=⨁n∈Z(PSL2(Qp),PSL2(Zp)). The cyclic group Z Z Z\mathbf{Z}Z naturally acts on M ( p ) M ( p ) M(p)M(p)M(p) by shifting the coordinates. The semidirect product G ( p ) = M ( p ) Z G ( p ) = M ( p ) ⋊ Z G(p)=M(p)><|ZG(p)=M(p) \rtimes \mathbf{Z}G(p)=M(p)⋊Z is a compactly generated tdlc group, with an essentially chief series given by { 1 } < M ( p ) < G ( p ) { 1 } < M ( p ) < G ( p ) {1} < M(p) < G(p)\{1\}<M(p)<G(p){1}<M(p)<G(p). The group M ( p ) M ( p ) M(p)M(p)M(p) is not compactly generated. It has minimal closed normal subgroups, but does not admit any finite essentially chief series, which illustrates the necessity of the compact generation hypothesis in Theorem 2.1.
Example 2.3. A more elaborate construction in [63, §9] yields an example of a compactly generated tdlc group G ( p ) G ′ ( p ) G^(')(p)G^{\prime}(p)G′(p) with an essentially chief series given by { 1 } < H ( p ) < G ( p ) { 1 } < H ( p ) < G ′ ( p ) {1} < H(p) < G^(')(p)\{1\}<H(p)<G^{\prime}(p){1}<H(p)<G′(p) such that G ( p ) / H ( p ) Z G ′ ( p ) / H ( p ) ≅ Z G^(')(p)//H(p)~=ZG^{\prime}(p) / H(p) \cong \mathbb{Z}G′(p)/H(p)≅Z and H ( p ) H ( p ) H(p)H(p)H(p) has a nested chain of closed normal subgroups ( H ( p ) n ) H ( p ) n (H(p)_(n))\left(H(p)_{n}\right)(H(p)n) indexed by Z Z Z\mathbf{Z}Z, permuted transitively by the conjugation G ( p ) G ′ ( p ) G^(')(p)G^{\prime}(p)G′(p)-action, and such that H ( p ) n / H ( p ) n 1 M ( p ) H ( p ) n / H ( p ) n − 1 ≅ M ( p ) H(p)_(n)//H(p)_(n-1)~=M(p)H(p)_{n} / H(p)_{n-1} \cong M(p)H(p)n/H(p)n−1≅M(p) for all n Z n ∈ Z n inZn \in \mathbf{Z}n∈Z.
A tdlc group is compactly generated if and only if it is capable of acting continuously, properly, with finitely many vertex orbits, by automorphisms on a connected locally finite graph. For a given compactly generated tdlc group G G GGG, vertex-transitive actions on graphs are afforded by the following construction. Given a compact open subgroup U < G U < G U < GU<GU<G, guaranteed to exist by van Dantzig's theorem, and a symmetric compact generating set Σ Î£ Sigma\SigmaΣ of G G GGG, we construct a graph Γ Î“ Gamma\GammaΓ whose vertex set is the coset space G / U G / U G//UG / UG/U by declaring that the vertices g U g U gUg UgU and h U h U hUh UhU are adjacent if h 1 g h − 1 g h^(-1)gh^{-1} gh−1g belongs to U Σ U U Σ U U Sigma UU \Sigma UUΣU. The fact that Σ Î£ Sigma\SigmaΣ generates G G GGG ensures that Γ Î“ Gamma\GammaΓ is connected. Moreover, G G GGG acts vertex-transitively by automorphisms on Γ Î“ Gamma\GammaΓ. Since U U UUU is compact open, the set U Σ U U Σ U U Sigma UU \Sigma UUΣU is a finite union of double cosets modulo U U UUU; this implies that Γ Î“ Gamma\GammaΓ is locally finite, i.e., the degree of each vertex is finite. Notice that all vertices have the same degree since Γ Î“ Gamma\GammaΓ is homogeneous. The graph Γ Î“ Gamma\GammaΓ is called a Cayley-Abels graph for G G GGG, since its construction was first envisaged by H. Abels [1, BEISPIEL 5.2] and specializes to a Cayley graph when G G GGG is discrete and U = { 1 } U = { 1 } U={1}U=\{1\}U={1}. The proof of Theorem 2.1 proceeds by induction on the minimum degree of a Cayley-Abels graph.

2.2. Elementary groups

By its very nature, Theorem 2.1 highlights the special role played by compact and discrete groups. A conceptual approach to studying the role of compact and discrete groups in the structure theory of tdlc groups is provided by P. Wesolek's notion of elementary groups. That concept is inspired by the class of elementary amenable discrete groups introduced by M. Day [33]. It is defined as the smallest class E E E\mathscr{E}E of second countable tdlc groups (abbreviated tdlcsc) containing all countable discrete groups and all compact tdlcsc groups, which is stable under the following two group theoretic operations:
  • Given a tdlcsc group G G GGG and a closed normal subgroup N N NNN, if N E N ∈ E N inEN \in \mathscr{E}N∈E and G / N E G / N ∈ E G//N inEG / N \in \mathscr{E}G/N∈E, then G E G ∈ E G inEG \in \mathscr{E}G∈E. In other words E E E\mathscr{E}E is stable under group extensions.
  • Given a tdlcsc group G G GGG and a directed set ( O i ) i I O i i ∈ I (O_(i))_(i in I)\left(O_{i}\right)_{i \in I}(Oi)i∈I of open subgroups, if O i E O i ∈ E O_(i)inEO_{i} \in \mathscr{E}Oi∈E for all i i iii and if G = i O i G = ⋃ i   O i G=uuu_(i)O_(i)G=\bigcup_{i} O_{i}G=⋃iOi, then G E G ∈ E G inEG \in \mathscr{E}G∈E. In other words E E E\mathscr{E}E is stable under directed unions of open subgroups.
(The class E E E\mathscr{E}E has a natural extension beyond the second countable case, see [29, $6]. For simplicity of the exposition, we stick to the second countable case here.) Using the permanence properties of the class E E E\mathscr{E}E, it can be shown that every tdlcsc group G G GGG has a largest closed normal subgroup that is elementary; it is denoted by R E ( G ) R E ( G ) R_(E)(G)R_{\mathscr{E}}(G)RE(G) and called the elementary radical of G G GGG. It indeed behaves as a radical, in the sense that it contains all elementary closed normal subgroups, and satisfies R E ( G / R E ( G ) ) = { 1 } R E G / R E ( G ) = { 1 } R_(E)(G//R_(E)(G))={1}R_{\mathscr{E}}\left(G / R_{\mathscr{E}}(G)\right)=\{1\}RE(G/RE(G))={1}, see [82, 87.2]. Further properties of the quotient G / R E ( G ) G / R E ( G ) G//R_(E)(G)G / R_{\mathscr{E}}(G)G/RE(G) will be mentioned in Section 3 below.
Similarly as for elementary amenable discrete groups, the class E E E\mathscr{E}E admits a canonical rank function ξ : E ω 1 ξ : E → ω 1 xi:Erarromega_(1)\xi: \mathscr{E} \rightarrow \omega_{1}ξ:E→ω1, taking values in the set ω 1 ω 1 omega_(1)\omega_{1}ω1 of countable ordinals, called the decomposition rank. It measures the complexity of a given group G E G ∈ E G inEG \in \mathscr{E}G∈E. By convention, the function ξ ξ xi\xiξ is extended to all tdlcsc groups by setting ξ ( G ) = ω 1 ξ ( G ) = ω 1 xi(G)=omega_(1)\xi(G)=\omega_{1}ξ(G)=ω1 for each nonelementary tdlcsc group G G GGG. We refer to [82], [83] and [62, $5]. Let us merely mention here that the class E E E\mathscr{E}E has remarkable permanence properties (e.g., it is stable under passing to closed subgroups and quotient groups), that the rank function has natural monotonicity properties, and that a nontrivial compactly generated group G E G ∈ E G inEG \in \mathscr{E}G∈E has a nontrivial discrete quotient. It follows in particular that if G G GGG is a tdlcsc group having a closed subgroup H G H ≤ G H <= GH \leq GH≤G admitting a nondiscrete compactly generated topologically simple quotient, then G E G ∉ E G!inEG \notin \mathscr{E}G∉E. Therefore, the only compactly generated topologically simple groups in E E E\mathscr{E}E are discrete. On the other hand, the class E E E\mathscr{E}E contains numerous topologically simple groups that are not compactly generated, e.g. simple groups that are regionally elliptic, i.e., groups that can be written as a directed union of compact open subgroups. Those groups have decomposition rank 2. Explicit examples appear in [ 88 , § 3 ] [ 88 , § 3 ] [88,§3][88, \S 3][88,§3] or [ 19 , § 6 ] [ 19 , § 6 ] [19,§6][19, \S 6][19,§6].

2.3. More on chief factors

The existence of essentially chief series prompts us to ask whether the chief factors of G G GGG depend upon the choice of a specific normal series in Theorem 2.1. It is tempting to tackle that question by invoking arguments à la Jordan-Hölder. A technical obstruction for doing so is that the product of two closed normal subgroups need not be closed More generally, given closed subgroups A , N A , N A,NA, NA,N in G G GGG such that N N NNN is normal, the product A N A N ANA NAN need not be closed so that the natural abstract isomorphism A / A N A N / N A / A ∩ N → A N / N A//A nn N rarr AN//NA / A \cap N \rightarrow A N / NA/A∩N→AN/N need not be a homeomorphism. It is a continuous injective homomorphism of the locally compact group A / A N A / A ∩ N A//A nn NA / A \cap NA/A∩N to a dense subgroup of the locally compact group A N ¯ / N A N ¯ / N bar(AN)//N\overline{A N} / NAN¯/N. This illustrates the necessity of considering dense embeddings of locally compact groups. We shall come back to this theme in Section 3.1 below. In the context of chief factors, this has led Reid-Wesolek to define an equivalence relation on nonabelian chief factors of G G GGG, called association, defined as follows: the chief factors K 1 / L 1 K 1 / L 1 K_(1)//L_(1)K_{1} / L_{1}K1/L1 and K 2 / L 2 K 2 / L 2 K_(2)//L_(2)K_{2} / L_{2}K2/L2 are associated if K 1 L 2 ¯ = K 2 L 1 ¯ K 1 L 2 ¯ = K 2 L 1 ¯ bar(K_(1)L_(2))= bar(K_(2)L_(1))\overline{K_{1} L_{2}}=\overline{K_{2} L_{1}}K1L2¯=K2L1¯ and K i L 1 L 2 ¯ = L i K i ∩ L 1 L 2 ¯ = L i K_(i)nn bar(L_(1)L_(2))=L_(i)K_{i} \cap \overline{L_{1} L_{2}}=L_{i}Ki∩L1L2¯=Li for i = 1 , 2 i = 1 , 2 i=1,2i=1,2i=1,2. In that case K 1 / L 1 K 1 / L 1 K_(1)//L_(1)K_{1} / L_{1}K1/L1 and K 2 / L 2 K 2 / L 2 K_(2)//L_(2)K_{2} / L_{2}K2/L2 both embed continuously as dense normal subgroups in K 1 K 2 ¯ / L 1 L 2 ¯ K 1 K 2 ¯ / L 1 L 2 ¯ bar(K_(1)K_(2))// bar(L_(1)L_(2))\overline{K_{1} K_{2}} / \overline{L_{1} L_{2}}K1K2¯/L1L2¯. We also recall that the quasicenter of a locally compact group G G GGG, denoted by Q Z ( G ) Q Z ( G ) QZ(G)\mathrm{QZ}(G)QZ(G), is the collection of elements whose centralizer is open. It is a topologically characteristic (not necessarily closed) subgroup of G G GGG containing all the discrete normal subgroups. It was first introduced
by M. Burger and S. Mozes [14]. Every nontrivial tdlcsc group with a dense quasicenter is elementary of decomposition rank 2 (see [62, LEM. 5]).
Theorem 2.4 (See [62, cor. 5]). Let G G GGG be a compactly generated tdlc group and let { 1 } = A 0 < A 1 < A 2 < < A m = G { 1 } = A 0 < A 1 < A 2 < ⋯ < A m = G {1}=A_(0) < A_(1) < A_(2) < cdots < A_(m)=G\{1\}=A_{0}<A_{1}<A_{2}<\cdots<A_{m}=G{1}=A0<A1<A2<⋯<Am=G and { 1 } = B 0 < B 1 < B 2 < < B n = G { 1 } = B 0 < B 1 < B 2 < ⋯ < B n = G {1}=B_(0) < B_(1) < B_(2) < cdots < B_(n)=G\{1\}=B_{0}<B_{1}<B_{2}<\cdots<B_{n}=G{1}=B0<B1<B2<⋯<Bn=G be essentially chief series for G G GGG. Then for each i { 0 , 1 , , m } i ∈ { 0 , 1 , … , m } i in{0,1,dots,m}i \in\{0,1, \ldots, m\}i∈{0,1,…,m}, if A i / A i 1 A i / A i − 1 A_(i)//A_(i-1)A_{i} / A_{i-1}Ai/Ai−1 is a chief factor with a trivial quasicenter, there is a unique j j jjj such that B j / B j 1 B j / B j − 1 B_(j)//B_(j-1)B_{j} / B_{j-1}Bj/Bj−1 is a chief factor with a trivial quasicenter that is associated with A i / A i 1 A i / A i − 1 A_(i)//A_(i-1)A_{i} / A_{i-1}Ai/Ai−1. In other words, the association relation establishes a bijection between the sets of chief factors with a trivial quasicenter appearing respectively in the two series.
The natural next question is to ask what can be said about chief factors. By the discussion above, one should focus on properties that are invariant under the association relation. Following Reid-Wesolek, an association class of nonabelian chief factors is called a chief block, and a group property shared by all members of a chief block is called a block property. The following are shown in [63] to be block properties: compact generation, amenability, having a trivial quasicenter, having a dense quasicenter, being elementary of a given decomposition rank.
As mentioned above, every chief factor is topologically characteristically simple. In particular, a compactly generated chief factor is subjected to the following description.
Theorem 2.5 (See [23, COR. D] and [22, REM. 3.10]). Let G be a compactly generated nondiscrete, noncompact tdlc group which is topologically characteristic simple. Then there is a compactly generated nondiscrete topologically simple tdlc group S S SSS, an integer d 1 d ≥ 1 d >= 1d \geq 1d≥1 and an injective continuous homomorphism S d = S × × S G S d = S × ⋯ × S → G S^(d)=S xx cdots xx S rarr GS^{d}=S \times \cdots \times S \rightarrow GSd=S×⋯×S→G of the direct product of d d ddd copies of S S SSS, such that the image of each simple factor is a closed normal subgroup of G G GGG, and the image of the whole product is dense.
In the setting of Theorem 2.5, we say that G G GGG is the quasiproduct d d ddd copies of the simple group S S SSS. Theorem 2.5 provides a major incentive to study the compactly generated nondiscrete topologically simple tdlc groups. We shall come back to this theme in Section 3 below.
Developing a meaningful structure theory for topologically characteristically simple tdlc groups that are not compactly generated is very challenging. Remarkably, significant results have been established by Reid-Wesolek [63] under the mild assumption of second countability (abbreviated sc). In spite of the noncompact generation, they introduce an appropriate notion of chief blocks, and show that there are only three possible configurations for the arrangement of chief blocks in a topologically characteristically simple tdlcsc group G G GGG, that they call weak type, semisimple type, and stacking type. Moreover, if G G GGG is of weak type, then it is automatically elementary of decomposition rank ω + 1 ≤ ω + 1 <= omega+1\leq \omega+1≤ω+1. The topologically characteristically simple groups M ( p ) M ( p ) M(p)M(p)M(p) and H ( p ) H ( p ) H(p)H(p)H(p) appearing in Examples 2.2 and 2.3 above are respectively of semisimple type and stacking type. We refer to [63] and [62] for details.

3. SIMPLE GROUPS

Let S S S\mathscr{S}S be the class of nondiscrete, compactly generated, topologically simple locally compact groups and S td S td  S_("td ")\mathscr{S}_{\text {td }}Std  be the subclass consisting of the totally disconnected members of S S S\mathscr{S}S. By the Gleason-Yamabe theorem [73, тн. 6.0.11], all elements of S S td S ∖ S td  S\\S_("td ")\mathscr{S} \backslash \mathscr{S}_{\text {td }}S∖Std  are connected simple Lie groups. Prominent examples of groups in S t d S t d S_(td)\mathscr{S}_{\mathrm{td}}Std are provided by simple algebraic groups over non-Archimedean local fields, irreducible complete Kac-Moody groups over finite fields, certain groups acting on trees and many more, see [28, APPENDIX A]. A systematic study of the class S td S td  S_("td ")\mathscr{S}_{\text {td }}Std  as a whole has been initiated by Caprace-Reid-Willis in [28], and continued with P. Wesolek in [29] and with A. Le Boudec in [22]. We now outline some of their contributions. Another survey of the properties of nondiscrete simple locally compact groups can be consulted in [17]; the present account emphasizes more recent results.

3.1. Dense embeddings and local structure

As mentioned in Section 2.3 above, the failure of the second isomorphism theorem for topological groups naturally leads one to consider dense embeddings, i.e., continuous injective homomorphisms with dense image. If G , H G , H G,HG, HG,H are locally compact groups and ψ ψ psi\psiψ : H G H → G H rarr GH \rightarrow GH→G is a dense embedding, and if G G GGG is a connected simple Lie group or a simple algebraic group over a local field, then H H HHH is discrete or ψ ψ psi\psiψ is an isomorphism (see [29, §3]). This property however generally fails for groups G S G ∈ S G inSG \in \mathscr{S}G∈S; see [50] for explicit examples. Nevertheless, as soon as the group H H HHH is nondiscrete, it turns out that key structural features of G G GGG are inherited by the dense subgroup H H HHH. To state this more precisely, we recall the definition of the class R R R\mathscr{R}R of robustly monolithic groups, introduced in [29]. A tdlc group G G GGG is robustly monolithic if the intersection M M MMM of all nontrivial closed normal subgroups of G G GGG is nontrivial, if M M MMM is topologically simple and if M M MMM has a compactly generated open subgroup without any nontrivial compact normal subgroup. The class R R R\mathscr{R}R contains S t d S t d S_(td)\mathscr{S}_{\mathrm{td}}Std and that inclusion is strict. The following result provides the main motivation to enlarge one's viewpoint by considering R R R\mathscr{R}R instead of the smaller class S t d S t d S_(td)\mathscr{S}_{\mathrm{td}}Std.
Theorem 3.1 (See [29, тн. 1.1.2]). Let G , H G , H G,HG, HG,H be tdlc groups and ψ : H G ψ : H → G psi:H rarr G\psi: H \rightarrow Gψ:H→G be a dense embedding. If G R G ∈ R G inRG \in \mathscr{R}G∈R and H H HHH is nondiscrete, then H R H ∈ R H inRH \in \mathscr{R}H∈R.
We emphasize that in general H H HHH is not topologically simple even in the special case where G S t d G ∈ S t d G inS_(td)G \in \mathscr{S}_{\mathrm{td}}G∈Std.
The approach in studying the classes S td S td  S_("td ")\mathscr{S}_{\text {td }}Std  and R R R\mathscr{R}R initiated in [28] is based on the concept of locally normal subgroup, defined as a subgroup whose normalizer is open. To motivate it, recall once more that if M , N M , N M,NM, NM,N are closed normal subgroups of a tdlc group G G GGG, then the normal subgroup M N M N MNM NMN need not be closed. On the other hand, if U G U ≤ G U <= GU \leq GU≤G is a compact open subgroup, then M U M ∩ U M nn UM \cap UM∩U and N U N ∩ U N nn UN \cap UN∩U are closed normal subgroups of the compact group U U UUU (hence they are both locally normal), so that the product ( M U ) ( N U ) ( M ∩ U ) ( N ∩ U ) (M nn U)(N nn U)(M \cap U)(N \cap U)(M∩U)(N∩U) is closed. This observation motivates the definition of the structure lattice L N ( G ) L N ( G ) LN(G)\mathscr{L} \mathcal{N}(G)LN(G) of a tdlc group G G GGG first introduced in [27], defined as the set of closed locally normal subgroups of G G GGG, divided by the local equivalence relation ∼ ∼\sim∼, where H K H ∼ K H∼KH \sim KH∼K if H K H ∩ K H nn KH \cap KH∩K is relatively open both in H H HHH
and in K K KKK. The local class of a closed locally normal subgroup K K KKK is denoted by [ K ] [ K ] [K][K][K]. We also set 0 = [ { 1 } ] 0 = [ { 1 } ] 0=[{1}]0=[\{1\}]0=[{1}] and = [ G ] ∞ = [ G ] oo=[G]\infty=[G]∞=[G]. The structure lattice carries a natural G G GGG-invariant order relation defined by the inclusion of representatives. The poset L N ( G ) L N ( G ) LN(G)\mathscr{L} \mathcal{N}(G)LN(G) is a modular lattice (see [27, LEM. 2.3]). The greatest lower bound and least upper bound of two elements α , β L N ( G ) α , β ∈ L N ( G ) alpha,beta inLN(G)\alpha, \beta \in \mathscr{L} \mathcal{N}(G)α,β∈LN(G) are respectively denoted by α β α ∧ β alpha^^beta\alpha \wedge \betaα∧β and α β α ∨ β alpha vv beta\alpha \vee \betaα∨β. When G G GGG is a p p ppp-adic Lie group, the structure lattice L N ( G ) L N ( G ) LN(G)\mathscr{L} \mathcal{N}(G)LN(G) can naturally be identified with the lattice of ideals in the Q p Q p Q_(p)\mathbf{Q}_{p}Qp-Lie algebra of G G GGG. The theory developed in [27] reveals that the structure lattice is especially well-behaved when the tdlc group G G GGG is [ A ] [ A ] [A][A][A]-semisimple, i.e., Q Z ( G ) = { 1 } Q Z ( G ) = { 1 } QZ(G)={1}\mathrm{QZ}(G)=\{1\}QZ(G)={1} and the only abelian locally normal subgroup of G G GGG is { 1 } { 1 } {1}\{1\}{1}. That term is motivated by the fact that if G G GGG is a p p ppp-adic Lie group, then it is [A]-semisimple if and only if Q Z ( G ) = { 1 } Q Z ( G ) = { 1 } QZ(G)={1}\mathrm{QZ}(G)=\{1\}QZ(G)={1} and the Q p Q p Q_(p)\mathbf{Q}_{p}Qp-Lie algebra of G G GGG is semisimple, see [27, PRoP. 6.18]. An important result of P. Wesolek is that the quotient G / R E ( G ) G / R E ( G ) G//R_(E)(G)G / R_{\mathscr{E}}(G)G/RE(G) of every tdlcsc group G G GGG by its elementary radical is [A]-semisimple (see [82, coR. 9.15]), so that every nonelementary group has a nontrivial [A]-semisimple quotient. The following result shows that [A]-semisimplicity is automatically fulfilled by groups in R R R\mathscr{R}R.
Theorem 3.2 (See [28, тн. A] and [29, тн. 1.2.5]). Every group G R G ∈ R G inRG \in \mathscr{R}G∈R is [A]-semisimple.
Given an [A]-semisimple tdlc group G G GGG, two closed locally normal subgroups H H HHH, K G K ≤ G K <= GK \leq GK≤G that are locally equivalent have the same centralizer; moreover, they commute if and only if their intersection is trivial (see [27, тн. 3.19]). This ensures that the map L N ( G ) L N ( G ) : [ K ] [ K ] = [ C G ( K ) ] L N ( G ) → L N ( G ) : [ K ] ↦ [ K ] ⊥ = C G ( K ) LN(G)rarrLN(G):[K]|->[K]^(_|_)=[C_(G)(K)]\mathscr{L} \mathcal{N}(G) \rightarrow \mathscr{L} \mathcal{N}(G):[K] \mapsto[K]^{\perp}=\left[C_{G}(K)\right]LN(G)→LN(G):[K]↦[K]⊥=[CG(K)] is well defined, and that α α = 0 α ∧ α ⊥ = 0 alpha^^alpha^(_|_)=0\alpha \wedge \alpha^{\perp}=0α∧α⊥=0 for all α L N ( G ) α ∈ L N ( G ) alpha inLN(G)\alpha \in \mathscr{L} \mathcal{N}(G)α∈LN(G). This allows one to define the centralizer lattice of G G GGG by setting L C ( G ) = L C ( G ) = LC(G)=\mathscr{L} \mathscr{C}(G)=LC(G)= { α α L N ( G ) } α ⊥ ∣ α ∈ L N ( G ) {alpha^(_|_)∣alpha inLN(G)}\left\{\alpha^{\perp} \mid \alpha \in \mathscr{L} \mathcal{N}(G)\right\}{α⊥∣α∈LN(G)}. If G G GGG is [A]-semisimple, the centralizer lattice L C ( G ) L C ( G ) LC(G)\mathscr{L} \mathscr{C}(G)LC(G) is a Boolean algebra (see [27, TH. II]). We denote its Stone dual by Ω G Ω G Omega_(G)\Omega_{G}ΩG. Thus Ω G Ω G Omega_(G)\Omega_{G}ΩG is a totally disconnected compact space endowed with a canonical continuous G G GGG-action by homeomorphisms. In general, the G G GGG-action on Ω G Ω G Omega_(G)\Omega_{G}ΩG need not be faithful. Actually, if L C ( G ) = { 0 , } L C ( G ) = { 0 , ∞ } LC(G)={0,oo}\mathscr{L} \mathscr{C}(G)=\{0, \infty\}LC(G)={0,∞} then Ω G Ω G Omega_(G)\Omega_{G}ΩG is a singleton. This happens if and only if any two non-trivial closed locally normal subgroups of G G GGG have a nontrivial intersection. The following result shows that the dynamics of the G G GGG-action on Ω G Ω G Omega_(G)\Omega_{G}ΩG has remarkable features.
Theorem 3.3 (See [28, тн. J] and [29, тн. 1.2.6]). Let G R G ∈ R G inRG \in \mathscr{R}G∈R. Then the G G GGG-action on Ω G Ω G Omega_(G)\Omega_{G}ΩG is minimal, strongly proximal, and has a compressible open set. Moreover, the G G GGG-action on Ω G Ω G Omega_(G)\Omega_{G}ΩG is faithful if and only if L C ( G ) { 0 , } L C ( G ) ≠ { 0 , ∞ } LC(G)!={0,oo}\mathscr{L} \mathscr{C}(G) \neq\{0, \infty\}LC(G)≠{0,∞}.
Recall that a compact G G GGG-space X X XXX is called minimal if every G G GGG-orbit is dense. It is called strongly proximal if the closure of each G G GGG-orbit in the space of probability measures on X X XXX contains a Dirac mass. A nonempty subset α α alpha\alphaα of X X XXX is called compressible if for every nonempty open subset β X β ⊆ X beta sube X\beta \subseteq Xβ⊆X there exists g G g ∈ G g in Gg \in Gg∈G with g α β g α ⊆ β g alpha sube betag \alpha \subseteq \betagα⊆β. Obviously, if X X XXX is a minimal strongly proximal compact G G GGG-space and if G G GGG fixes a probability measure on X X XXX, then X X XXX is a singleton. Therefore, the following consequence of Theorem 3.3 is immediate.
Corollary 3.4. Let G R G ∈ R G inRG \in \mathscr{R}G∈R. If G G GGG is amenable, then L C ( G ) = { 0 , } L C ( G ) = { 0 , ∞ } LC(G)={0,oo}\mathscr{L} \mathscr{C}(G)=\{0, \infty\}LC(G)={0,∞}.
A local isomorphism between tdlc groups G 1 , G 2 G 1 , G 2 G_(1),G_(2)G_{1}, G_{2}G1,G2 is a triple ( φ , U 1 , U 2 φ , U 1 , U 2 varphi,U_(1),U_(2)\varphi, U_{1}, U_{2}φ,U1,U2 ) where U i U i U_(i)U_{i}Ui is an open subgroup of G i G i G_(i)G_{i}Gi and φ : U 1 U 2 φ : U 1 → U 2 varphi:U_(1)rarrU_(2)\varphi: U_{1} \rightarrow U_{2}φ:U1→U2 is an isomorphism of topological groups. We
emphasize that the structure lattice and the centralizer lattice are local invariants: they only depend on the local isomorphism class of the ambient tdlc group. However, for a group G R G ∈ R G inRG \in \mathscr{R}G∈R, the compact G G GGG-space Ω G Ω G Omega_(G)\Omega_{G}ΩG can also be characterized by global properties among all compact G G GGG-spaces. In order to be more precise, let us first recall some terminology. Given an action of a group G G GGG by homeomorphisms on a Hausdorff topological space X X XXX, we define the rigid stabilizer Rist G ( U ) Rist G ⁡ ( U ) Rist_(G)(U)\operatorname{Rist}_{G}(U)RistG⁡(U) of a subset U X U ⊆ X U sube XU \subseteq XU⊆X as the pointwise stabilizer of the complement of U U UUU in X X XXX. The G G GGG-action on X X XXX is called microsupported if for every nonempty open subset U X U ⊂ X U sub XU \subset XU⊂X with U X U ≠ X U!=XU \neq XU≠X, the rigid stabilizer Rist G ( U ) Rist G ⁡ ( U ) Rist_(G)(U)\operatorname{Rist}_{G}(U)RistG⁡(U) acts nontrivially on X X XXX. The term "microsupported" was first coined in [28], although the notion it designates has frequently appeared in earlier references, notably in the work of M. Rubin on reconstruction theorems (see [67] and references therein). A prototypical example of a microsupported action of a tdlc group is given by the action of the full automorphism group Aut ( T ) Aut ⁡ ( T ) Aut(T)\operatorname{Aut}(T)Aut⁡(T) of a locally finite regular tree T T TTT of degree 3 ≥ 3 >= 3\geq 3≥3 on the compact space T ∂ T del T\partial T∂T consisting of the ends of T T TTT. The following result shows that for a general group G R G ∈ R G inRG \in \mathscr{R}G∈R, the G G GGG-action on Ω G Ω G Omega_(G)\Omega_{G}ΩG shares many dynamical properties with the Aut ( T ) Aut ⁡ ( T ) Aut(T)\operatorname{Aut}(T)Aut⁡(T)-action on T ∂ T del T\partial T∂T.
Theorem 3.5 (See [28, тн. J], [29, тн. 7.3.3] and [22, тн. 7.5]). Let G R G ∈ R G inRG \in \mathscr{R}G∈R. Then the G-action on Ω G Ω G Omega_(G)\Omega_{G}ΩG is microsupported. Moreover, for each nonempty microsupported compact G G GGG-space X X XXX on which the G G GGG-action is faithful, there is a G-equivariant continuous surjective map Ω G X Ω G → X Omega_(G)rarr X\Omega_{G} \rightarrow XΩG→X. In particular, the G G GGG-action on X X XXX is minimal, strongly proximal, and has a compressible open set.
This shows that Ω G Ω G Omega_(G)\Omega_{G}ΩG is universal among the faithful microsupported compact G G GGG-spaces; in particular, the purely local condition that L C ( G ) = { 0 , } L C ( G ) = { 0 , ∞ } LC(G)={0,oo}\mathscr{L} \mathscr{C}(G)=\{0, \infty\}LC(G)={0,∞} ensures that G G GGG does not have any faithful microsupported continuous action on any compact space. Theorem 3.5 was first established for totally disconnected compact G G GGG-spaces in [28,29], and then extended to all compact G G GGG-spaces in [22], using tools from topological dynamics. Further properties of the G G GGG-space Ω G Ω G Omega_(G)\Omega_{G}ΩG and on the algebraic structure of groups in R R R\mathscr{R}R can be consulted in those references.
We now present another aspect of the local approach to the structure of simple tdlc groups. We define the local prime content of a tdlc group G G GGG, denoted by π ( G ) Ï€ ( G ) pi(G)\pi(G)Ï€(G), to be the set of those primes p p ppp such that every compact open subgroup U G U ≤ G U <= GU \leq GU≤G contains an infinite pro- p p ppp subgroup.
Theorem 3.6 (See [28, TH. H] and [29, coR. 1.1.4 AND TH. 1.2.1]). The following assertions hold for any group G R G ∈ R G inRG \in \mathscr{R}G∈R :
(i) The local prime content π ( G ) Ï€ ( G ) pi(G)\pi(G)Ï€(G) is finite and nonempty.
(ii) For each p π ( G ) p ∈ Ï€ ( G ) p in pi(G)p \in \pi(G)p∈π(G), there is a group G ( p ) R G ( p ) ∈ R G_((p))inRG_{(p)} \in \mathscr{R}G(p)∈R that is locally isomorphic to a pro-p group, and a dense embedding G ( p ) G G ( p ) → G G_((p))rarr GG_{(p)} \rightarrow GG(p)→G.
(iii) If H H HHH is a tdlc group acting continuously and faithfully by automorphisms on G G GGG, then H H HHH is locally isomorphic to a pro- π ( G ) Ï€ ( G ) pi(G)\pi(G)Ï€(G) group.
Roughly speaking, Theorem 3.6(ii) asserts that every group in R R R\mathscr{R}R can be "approximated" by a locally pro- p p ppp group in R R R\mathscr{R}R. The restriction on the automorphism group of a group in R R R\mathscr{R}R from Theorem 3.6(iii) should be compared with the automorphism group of the restricted product M ( p ) M ( p ) M(p)M(p)M(p) from Example 2.2. Indeed, the Polish group Sym ( Z ) Sym ⁡ ( Z ) Sym(Z)\operatorname{Sym}(\mathbf{Z})Sym⁡(Z) embeds continuously in Aut ( M ( p ) ) Aut ⁡ ( M ( p ) ) Aut(M(p))\operatorname{Aut}(M(p))Aut⁡(M(p)) by permuting the simple factors, and every tdlcsc group continuously embeds in Sym ( Z ) Sym ⁡ ( Z ) Sym(Z)\operatorname{Sym}(\mathbf{Z})Sym⁡(Z). In some sense, the construction of stacking type chief factors in Example 2.3 crucially relies on the hugeness of the group Aut ( M ( p ) ) Aut ⁡ ( M ( p ) ) Aut(M(p))\operatorname{Aut}(M(p))Aut⁡(M(p)). Theorem 3.6(iii) shows that the automorphism group of a group in R R R\mathscr{R}R is considerably smaller.
Let us finish this subsection with a brief discussion of classification problems. The work of S. Smith [72] shows that S td S td  S_("td ")\mathscr{S}_{\text {td }}Std  contains uncountably many isomorphism classes; his methods of proof suggest that the isomorphism relation on S td S td  S_("td ")\mathscr{S}_{\text {td }}Std  has a similar complexity as the isomorphism relation on the class of finitely generated discrete simple groups. This provides evidence that the problem of classifying groups in S td S td  S_("td ")\mathscr{S}_{\text {td }}Std  up to isomorphism is illposed. The recent results on the local structure of groups in S td S td  S_("td ")\mathscr{S}_{\text {td }}Std  or in R R R\mathscr{R}R may be viewed as a hint to the fact the local isomorphism relation might be better behaved (see [29, тн. 1.1.5]). At the time of this writing, we do not know whether or not the groups in S td S td  S_("td ")\mathscr{S}_{\text {td }}Std  fall into countably many local isomorphism classes. However, classifying simple groups up to isomorphism remains a pertinent problem for some significant subclasses of S td S td  S_("td ")\mathscr{S}_{\text {td }}Std . To wit, let us mention that, by [30, coR. 1.4], a group G S td G ∈ S td  G inS_("td ")G \in \mathscr{S}_{\text {td }}G∈Std  is isomorphic to a simple algebraic group over a local field if and only if it is locally isomorphic to a linear group, i.e., a subgroup of G L d ( k ) G L d ( k ) GL_(d)(k)\mathrm{GL}_{d}(k)GLd(k) for some integer d d ddd and some locally compact field k k kkk. Lastly, a remarkable classification theorem concerning an important class of nonlinear simple groups acting on locally finite trees has been obtained by N. Radu [61]. It would be highly interesting to extend Radu's results by classifying all groups in S td S td  S_("td ")\mathscr{S}_{\text {td }}Std  acting properly and continuously by automorphisms on a given locally finite tree T T TTT in such a way that the action on the set of ends of T T TTT is doubly transitive. That class is denoted by S T S T S_(T)\mathscr{S}_{T}ST. Results from [25] ensure that the isomorphism relation restricted to S T S T S_(T)\mathscr{S}_{T}ST is smooth (see [37, DEFINITION 5.4.1]), which means that it comes at the bottom of the hierarchy of complexity of classification problems in the formalism established by invariant

not know whether there is a tree T T TTT such that S T S T S_(T)\mathscr{S}_{T}ST contains uncountably many isomorphism classes.

3.2. Applications to lattices

The study of lattices in semisimple Lie and algebraic groups has known tremendous developments since the mid-20th century, with Margulis' seminal contributions as cornerstones. Remarkably, several key results on lattices have been established at a high level of generality, well beyond the realm of linear groups. An early illustration is provided by [13]. More recently, Y. Shalom [70] and Bader-Shalom [5] have established an extension of Margulis' Normal Subgroup Theorem valid for all irreducible cocompact lattices in products of groups in S S S\mathscr{S}S, while various analogues of Margulis' superrigidity for irreducible lattices in products have been established for various kinds of target spaces, see [ 3 , 4 , 31 , 36 , 38 , 55 , 56 , 70 ] [ 3 , 4 , 31 , 36 , 38 , 55 , 56 , 70 ] [3,4,31,36,38,55,56,70][3,4,31,36,38,55,56,70][3,4,31,36,38,55,56,70]. Those results have in common that they rely on transcendental methods: they use a mix
of tools from ergodic theory, probability theory, and abstract harmonic analysis, but do not require any detailed consideration of the algebraic structure of the ambient group. Another breakthrough in this field was accomplished by M. Burger and S. Mozes [15], who constructed a broad family of new finitely presented infinite simple groups as irreducible lattices in products of nonlinear groups in S td S td  S_("td ")\mathscr{S}_{\text {td }}Std . Their seminal work involves a mix of transcendental methods together with a fair amount of structure theory developed in [14].
The following two recent results rely in an essential way on the properties of the class S t d S t d S_(td)\mathscr{S}_{\mathrm{td}}Std outlined above.
Theorem 3.7 (See [21, тн. A]). Let n 2 n ≥ 2 n >= 2n \geq 2n≥2 be an integer, let G 1 , , G n S td G 1 , … , G n ∈ S td  G_(1),dots,G_(n)inS_("td ")G_{1}, \ldots, G_{n} \in \mathscr{S}_{\text {td }}G1,…,Gn∈Std  and Γ G = Γ ≤ G = Gamma <= G=\Gamma \leq G=Γ≤G= G 1 × × G n G 1 × ⋯ × G n G_(1)xx cdots xxG_(n)G_{1} \times \cdots \times G_{n}G1×⋯×Gn be a lattice such that the projection p i ( Γ ) p i ( Γ ) p_(i)(Gamma)p_{i}(\Gamma)pi(Γ) is dense in G i G i G_(i)G_{i}Gi for all i i iii. Assume that Γ Î“ Gamma\GammaΓ is cocompact, or that G G GGG has Kazhdan's property ( T ) ( T ) (T)(T)(T). Then the set of discrete subgroups of G G GGG containing Γ Î“ Gamma\GammaΓ is finite.
Theorem 3.8 (See [21, тн. c]). Let n 2 n ≥ 2 n >= 2n \geq 2n≥2 be an integer and let G 1 , , G n S t d G 1 , … , G n ∈ S t d G_(1),dots,G_(n)inS_(td)G_{1}, \ldots, G_{n} \in \mathscr{S}_{\mathrm{td}}G1,…,Gn∈Std be compactly presented. For every compact subset K G = G 1 × × G n K ⊂ G = G 1 × ⋯ × G n K sub G=G_(1)xx cdots xxG_(n)K \subset G=G_{1} \times \cdots \times G_{n}K⊂G=G1×⋯×Gn, the set of discrete subgroups Γ G Γ ≤ G Gamma <= G\Gamma \leq GΓ≤G with G = K Γ G = K Γ G=K GammaG=K \GammaG=KΓ and with p i ( Γ ) p i ( Γ ) p_(i)(Gamma)p_{i}(\Gamma)pi(Γ) dense in G i G i G_(i)G_{i}Gi for all i i iii, is contained in a union of finitely many Aut ( G ) Aut ⁡ ( G ) Aut(G)\operatorname{Aut}(G)Aut⁡(G)-orbits.
For a detailed discussion of the notion of compactly presented locally compact groups, we refer to [ 32 , c H .8 [ 32 , c H .8 [32,cH.8[32, \mathbf{c H} .8[32,cH.8.
Theorems 3.7 and 3.8 can be viewed as respective analogues of two theorems of H. C. Wang [ 78 , 79 ] [ 78 , 79 ] [78,79][78,79][78,79] on lattices in semisimple Lie groups and reveal the existence of positive lower bounds on the covolume of certain families of irreducible cocompact lattices. It should be underlined that the corresponding statements fail for lattices in a single group G S t d G ∈ S t d G inS_(td)G \in \mathscr{S}_{\mathrm{td}}G∈Std, see [6, тн. 7.1]. Theorem 3.8 is established by combining Theorem 3.7 with recent results on local rigidity of cocompact lattices in arbitrary groups, due to Gelander-Levit [39].

3.3. Applications to commensurated subgroups

The structure theory of tdlc groups provides valuable tools in exploring the so-called commensurated subgroups of an abstract group. In this section, we recall that connection and illustrate it with several recent results. Further results on commensurated subgroups will be mentioned in Section 4 below.
Let Γ Î“ Gamma\GammaΓ be a group. Two subgroups Λ 1 , Λ 2 Γ Î› 1 , Λ 2 ≤ Γ Lambda_(1),Lambda_(2) <= Gamma\Lambda_{1}, \Lambda_{2} \leq \GammaΛ1,Λ2≤Γ are called commensurate if their intersection Λ 1 Λ 2 Λ 1 ∩ Λ 2 Lambda_(1)nnLambda_(2)\Lambda_{1} \cap \Lambda_{2}Λ1∩Λ2 has finite index both in Λ 1 Λ 1 Lambda_(1)\Lambda_{1}Λ1 and in Λ 2 Λ 2 Lambda_(2)\Lambda_{2}Λ2. The commensurator of a subgroup Λ Γ Î› ≤ Γ Lambda <= Gamma\Lambda \leq \GammaΛ≤Γ, denoted by Comm Γ ( Λ ) Comm Γ ⁡ ( Λ ) Comm_(Gamma)(Lambda)\operatorname{Comm}_{\Gamma}(\Lambda)CommΓ⁡(Λ), is the set of those γ Γ Î³ ∈ Γ gamma in Gamma\gamma \in \Gammaγ∈Γ such that Λ Î› Lambda\LambdaΛ and γ Λ γ 1 γ Λ γ − 1 gamma Lambdagamma^(-1)\gamma \Lambda \gamma^{-1}γΛγ−1 are commensurate. It is easy to see that Comm Γ ( Λ ) Comm Γ ⁡ ( Λ ) Comm_(Gamma)(Lambda)\operatorname{Comm}_{\Gamma}(\Lambda)CommΓ⁡(Λ) is a subgroup of Γ Î“ Gamma\GammaΓ containing the normalizer N Γ ( Λ ) N Γ ( Λ ) N_(Gamma)(Lambda)N_{\Gamma}(\Lambda)NΓ(Λ). The commensurator has naturally appeared in group theory; one of its early occurrences is in Mackey's irreducibility criterion for induced unitary representations (see [51]). It also appears in a celebrated characterization of arithmetic lattices in semisimple groups due to Margulis [53, сH. I x , T H I x , T H Ix,TH\mathbf{I x}, \mathbf{T H}Ix,TH. (B)]. A commensurated subgroup of Γ Î“ Gamma\GammaΓ is a subgroup Λ Γ Î› ≤ Γ Lambda <= Gamma\Lambda \leq \GammaΛ≤Γ such that Comm Γ ( Λ ) = Γ Comm Γ ⁡ ( Λ ) = Γ Comm_(Gamma)(Lambda)=Gamma\operatorname{Comm}_{\Gamma}(\Lambda)=\GammaCommΓ⁡(Λ)=Γ. Clearly, every normal subgroup of Γ Î“ Gamma\GammaΓ is commensurated; more generally, every subgroup that is commensurate to a normal subgroup is commensurated.
Those commensurated subgroups are considered as trivial. For example, finite subgroups and subgroups of finite index are always commensurated subgroups. It is however important to underline that commensurated subgroups are not all of this trivial form. Indeed, an easy but crucial observation is that compact open subgroups are always commensurated. In particular, in the simple group P S L 2 ( Q p ) P S L 2 Q p PSL_(2)(Q_(p))\mathrm{PSL}_{2}\left(\mathbf{Q}_{p}\right)PSL2(Qp), the subgroup P S L 2 ( Z p ) P S L 2 Z p PSL_(2)(Z_(p))\mathrm{PSL}_{2}\left(\mathbf{Z}_{p}\right)PSL2(Zp) (which is obviously not commensurate to any normal subgroup of P S L 2 ( Q p ) P S L 2 Q p PSL_(2)(Q_(p))\mathrm{PSL}_{2}\left(\mathbf{Q}_{p}\right)PSL2(Qp) ) is commensurated.
Let us next remark that if U U UUU is a commensurated subgroup of a group G G GGG and φ : Γ G φ : Γ → G varphi:Gamma rarr G\varphi: \Gamma \rightarrow Gφ:Γ→G is a group homomorphism, then φ 1 ( U ) φ − 1 ( U ) varphi^(-1)(U)\varphi^{-1}(U)φ−1(U) is a commensurated subgroup of Γ Î“ Gamma\GammaΓ. This is the case in particular if G G GGG is a tdlc group and U G U ≤ G U <= GU \leq GU≤G is a compact open subgroup. A fundamental observation is that all commensurated subgroups of Γ Î“ Gamma\GammaΓ arise in this way. More, precisely, a subgroup Λ Γ Î› ≤ Γ Lambda <= Gamma\Lambda \leq \GammaΛ≤Γ is commensurated if and only if there is a tdlc group G G GGG, a compact open subgroup U G U ≤ G U <= GU \leq GU≤G, and a homomorphism φ : Γ G φ : Γ → G varphi:Gamma rarr G\varphi: \Gamma \rightarrow Gφ:Γ→G with dense image such that φ 1 ( U ) = Λ Ï† − 1 ( U ) = Λ varphi^(-1)(U)=Lambda\varphi^{-1}(U)=\Lambdaφ−1(U)=Λ. Indeed, given a commensurated subgroup Λ Γ Î› ≤ Γ Lambda <= Gamma\Lambda \leq \GammaΛ≤Γ, then Λ Î› Lambda\LambdaΛ acts on the coset space Γ / Λ Î“ / Λ Gamma//Lambda\Gamma / \LambdaΓ/Λ with finite orbits, so that the closure of the natural image of Γ Î“ Gamma\GammaΓ in the permutation group Sym ( Γ / Λ ) Sym ⁡ ( Γ / Λ ) Sym(Gamma//Lambda)\operatorname{Sym}(\Gamma / \Lambda)Sym⁡(Γ/Λ), endowed with the topology of pointwise convergence, is a tdlc group containing the closure of the image of Λ Î› Lambda\LambdaΛ as a compact open subgroup. That tdlc group is called the Schlichting completion of the pair ( Γ , Λ ) ( Γ , Λ ) (Gamma,Lambda)(\Gamma, \Lambda)(Γ,Λ), denoted by Γ / / Λ Î“ / / Λ Gamma////Lambda\Gamma / / \LambdaΓ//Λ. We refer to [68], [71, SECTION 3] and [65] for more information. Let us merely mention here that a commensurated subgroup Λ Γ Î› ≤ Γ Lambda <= Gamma\Lambda \leq \GammaΛ≤Γ is commensurate to a normal subgroup if and only if the Schlichting completion G = Γ / / Λ G = Γ / / Λ G=Gamma////LambdaG=\Gamma / / \LambdaG=Γ//Λ is compact-by-discrete, i.e., G G GGG has a compact open normal subgroup (see [22, LEM. 5.1]).
The occurrence of nontrivial commensurated subgroups in finitely generated groups with few normal subgroups (e.g., simple groups, or just-infinite groups, i.e., groups all of whose proper quotients are finite) remains an intriguing phenomenon. On the empirical basis of the known examples, it seems to be rather rare. The following result provides valuable information in that context.
Theorem 3.9 (See [22, тн. 5.4]). Let Γ Î“ Gamma\GammaΓ be a finitely generated group. Assume that all normal subgroups of Γ Î“ Gamma\GammaΓ are finitely generated, and that every proper quotient of Γ Î“ Gamma\GammaΓ is virtually nilpotent. Let also X X XXX be a compact Γ Î“ Gamma\GammaΓ-space on which the Γ Î“ Gamma\GammaΓ-action is faithful, minimal and microsupported. Assume that at least one of the following conditions is satisfied:
(1) Γ Î“ Gamma\GammaΓ is residually finite.
(2) Γ Î“ Gamma\GammaΓ fixes a probability measure on X X XXX.
Then every commensurated subgroup of Γ Î“ Gamma\GammaΓ is commensurate to a normal subgroup.
This applies to all finitely generated branch groups, as well as to numerous finitely generated almost simple groups arising in Cantor dynamics, and whose study has known spectacular recent developments (see [34,59] and references therein). We refer to [22] for details and a more precise description of those applications.
Let us briefly outline how the proof of Theorem 3.9 works in the case where Γ Î“ Gamma\GammaΓ fixes a probability measure on X X XXX. Let Λ Γ Î› ≤ Γ Lambda <= Gamma\Lambda \leq \GammaΛ≤Γ be a commensurated subgroup and G = Γ / / Λ G = Γ / / Λ G=Gamma////LambdaG=\Gamma / / \LambdaG=Γ//Λ be the
corresponding Schlichting completion. That Γ Î“ Gamma\GammaΓ is finitely generated implies that G G GGG is compactly generated. The hypotheses made on the normal subgroup structure of Γ Î“ Gamma\GammaΓ yield some restrictions on the essentially chief series of G G GGG afforded by Theorem 2.1. More precisely, assuming by contradiction that Λ Î› Lambda\LambdaΛ is not commensurate to a normal subgroup, then the upper most chief factor K / L K / L K//LK / LK/L with trivial quasicenter in an essentially chief series for G G GGG must be compactly generated. Its structure is therefore described by Theorem 2.5. A key point in the proof, relying on various ingredients from topological dynamics and involving detailed considerations of the Chabauty space of closed subgroups of Γ Î“ Gamma\GammaΓ and G G GGG, is to show that the given Γ Î“ Gamma\GammaΓ-action on X X XXX gives rise to a continuous, faithful, microsupported G / L G / L G//LG / LG/L-action on a compact space Y Y YYY which is closely related to the original space X X XXX. Invoking (a suitable version of) Theorem 3.5 for the chief factor K / L K / L K//LK / LK/L ensures that Y Y YYY has a compressible open set, from which it follows that X X XXX has a compressible open set for the Γ Î“ Gamma\GammaΓ-action. This finally contradicts the hypothesis of existence of a Γ Î“ Gamma\GammaΓ-invariant probability measure.

4. SCALE METHODS

The scale of a tdlc group endomorphism, α α alpha\alphaα, is a positive integer that conveys information about the dynamics of the action of α α alpha\alphaα. Roughly speaking, α α alpha\alphaα contracts towards the identity on one subgroup of G G GGG and expands on another, and the scale is the expansion factor This section gives an account of properties of the scale and descriptions of the action of α α alpha\alphaα on certain associated subgroups of G G GGG which, when applied to inner automorphisms, answer questions about group structure.
Let α : G G α : G → G alpha:G rarr G\alpha: G \rightarrow Gα:G→G be a continuous endomorphism. The scale of α α alpha\alphaα is
s ( α ) = min { [ α ( U ) : α ( U ) U ] U G compact and open } s ( α ) = min { [ α ( U ) : α ( U ) ∩ U ] ∣ U ≤ G  compact and open  } s(alpha)=min{[alpha(U):alpha(U)nn U]∣U <= G" compact and open "}s(\alpha)=\min \{[\alpha(U): \alpha(U) \cap U] \mid U \leq G \text { compact and open }\}s(α)=min{[α(U):α(U)∩U]∣U≤G compact and open }
This value is a positive integer because α ( U ) U α ( U ) ∩ U alpha(U)nn U\alpha(U) \cap Uα(U)∩U is an open subgroup of the compact group α ( U ) α ( U ) alpha(U)\alpha(U)α(U). Subgroups at which the minimum is attained are said to be minimizing for α α alpha\alphaα. The following results from [ 85 , 86 , 89 ] [ 85 , 86 , 89 ] [85,86,89][85,86,89][85,86,89] relate minimizing subgroups to the dynamics of α α alpha\alphaα.
Theorem 4.1. Let α α alpha\alphaα be a continuous endomorphism of the tdlc group G G GGG and let U G U ≤ G U <= GU \leq GU≤G be compact and open. Define subgroups
U + = { u U { u n } n 0 U with u 0 = u and u n = α ( u n + 1 ) } U = { u U α n ( u ) U for all n 0 } . U + = u ∈ U ∣ ∃ u n n ≥ 0 ⊂ U  with  u 0 = u  and  u n = α u n + 1 U − = u ∈ U ∣ α n ( u ) ∈ U  for all  n ≥ 0 . {:[U_(+)={u in U∣EE{u_(n)}_(n >= 0)sub U" with "u_(0)=u" and "u_(n)=alpha(u_(n+1))}],[U_(-)={u in U∣alpha^(n)(u)in U" for all "n >= 0}.]:}\begin{aligned} & U_{+}=\left\{u \in U \mid \exists\left\{u_{n}\right\}_{n \geq 0} \subset U \text { with } u_{0}=u \text { and } u_{n}=\alpha\left(u_{n+1}\right)\right\} \\ & U_{-}=\left\{u \in U \mid \alpha^{n}(u) \in U \text { for all } n \geq 0\right\} . \end{aligned}U+={u∈U∣∃{un}n≥0⊂U with u0=u and un=α(un+1)}U−={u∈U∣αn(u)∈U for all n≥0}.
Also define the subgroup U = n 0 α n ( U ) U − − = ⋃ n ≥ 0   α − n U − U_(--)=uuu_(n >= 0)alpha^(-n)(U_(-))U_{--}=\bigcup_{n \geq 0} \alpha^{-n}\left(U_{-}\right)U−−=⋃n≥0α−n(U−)of G.
Then U U UUU is minimizing for α α alpha\alphaα if and only if
(TA) U = U + U U = U + U − U=U_(+)U_(-)U=U_{+} U_{-}U=U+U−and (TB) U U − U_(-)U_{-}U−is closed.
A compact open subgroup U U UUU satisfying T A T A TAT ATA and T B T B TBT BTB is said to be tidy for α α alpha\alphaα, and s ( α ) = [ α ( U + ) : U + ] s ( α ) = α U + : U + s(alpha)=[alpha(U_(+)):U_(+)]s(\alpha)=\left[\alpha\left(U_{+}\right): U_{+}\right]s(α)=[α(U+):U+]for any such subgroup U U UUU. Tidiness has two further dynamical interpretations: (1) an α α alpha\alphaα-trajectory { α n ( g ) } n 0 α n ( g ) n ≥ 0 {alpha^(n)(g)}_(n >= 0)\left\{\alpha^{n}(g)\right\}_{n \geq 0}{αn(g)}n≥0 cannot return to a tidy subgroup once it departs; and (2) when α α alpha\alphaα is an automorphism, U U UUU is tidy for α α alpha\alphaα if and only if the orbit { α n ( U ) } n Z α n ( U ) n ∈ Z {alpha^(n)(U)}_(n inZ)\left\{\alpha^{n}(U)\right\}_{n \in \mathbb{Z}}{αn(U)}n∈Z is a
geodesic for the metric d ( U , V ) = log [ U : U V ] + log [ V : U V ] d ( U , V ) = log ⁡ [ U : U ∩ V ] + log ⁡ [ V : U ∩ V ] d(U,V)=log[U:U nn V]+log[V:U nn V]d(U, V)=\log [U: U \cap V]+\log [V: U \cap V]d(U,V)=log⁡[U:U∩V]+log⁡[V:U∩V] on the set of compact open subgroups of G G GGG.
Note that every compact open subgroup of G G GGG has a subgroup U U UUU for which T A T A TAT ATA holds and, if α α alpha\alphaα is the inner automorphism α g ( x ) := g x g 1 α g ( x ) := g x g − 1 alpha_(g)(x):=gxg^(-1)\alpha_{g}(x):=g x g^{-1}αg(x):=gxg−1, then property T A T A TAT ATA implies that U g m U g n U = U g m + n U U g m U g n U = U g m + n U Ug^(m)Ug^(n)U=Ug^(m+n)UU g^{m} U g^{n} U=U g^{m+n} UUgmUgnU=Ugm+nU for all m , n 0 m , n ≥ 0 m,n >= 0m, n \geq 0m,n≥0. These points were already used in [12] in the proof that a reductive group over a locally compact field of positive characteristic is type I, where they were observed to hold in such groups.
In the following compilation of results from [ 54 , 85 , 86 , 89 ] , Δ [ 54 , 85 , 86 , 89 ] , Δ [54,85,86,89],Delta[54,85,86,89], \Delta[54,85,86,89],Δ denotes the modular function on the automorphism group of G G GGG.
Theorem 4.2. The scale s : End ( G ) Z + s : End ⁡ ( G ) → Z + s:End(G)rarrZ^(+)s: \operatorname{End}(G) \rightarrow \mathbb{Z}^{+}s:End⁡(G)→Z+satisfies:
(i) s ( α ) = 1 s ( α ) = 1 quad s(alpha)=1\quad s(\alpha)=1s(α)=1 if and only if there is a compact open subgroup U G U ≤ G U <= GU \leq GU≤G with α ( U ) U α ( U ) ≤ U alpha(U) <= U\alpha(U) \leq Uα(U)≤U;
(ii) s ( α ) = lim n [ α n ( V ) : α n ( V ) V ] 1 n s ( α ) = lim n → ∞   α n ( V ) : α n ( V ) ∩ V 1 n s(alpha)=lim_(n rarr oo)[alpha^(n)(V):alpha^(n)(V)nn V]^((1)/(n))s(\alpha)=\lim _{n \rightarrow \infty}\left[\alpha^{n}(V): \alpha^{n}(V) \cap V\right]^{\frac{1}{n}}s(α)=limn→∞[αn(V):αn(V)∩V]1n for every compact open V G V ≤ G V <= GV \leq GV≤G, and s ( α n ) = s ( α ) n s α n = s ( α ) n s(alpha^(n))=s(alpha)^(n)s\left(\alpha^{n}\right)=s(\alpha)^{n}s(αn)=s(α)n for every n 0 n ≥ 0 n >= 0n \geq 0n≥0; and
(iii) if α α alpha\alphaα is an automorphism, then Δ ( α ) = s ( α ) / s ( α 1 ) Δ ( α ) = s ( α ) / s α − 1 Delta(alpha)=s(alpha)//s(alpha^(-1))\Delta(\alpha)=s(\alpha) / s\left(\alpha^{-1}\right)Δ(α)=s(α)/s(α−1).
The function s α : G Z + s ∘ α ∙ : G → Z + s@alpha_(∙):G rarrZ^(+)s \circ \alpha_{\bullet}: G \rightarrow \mathbb{Z}^{+}s∘α∙:G→Z+, with α g ( x ) = g x g 1 α g ( x ) = g x g − 1 alpha_(g)(x)=gxg^(-1)\alpha_{g}(x)=g x g^{-1}αg(x)=gxg−1, is continuous for the group topology on G G GGG and the discrete topology on Z + Z + Z^(+)\mathbb{Z}^{+}Z+.
Continuity of s α s ∘ α ∙ s@alpha_(∙)s \circ \alpha_{\bullet}s∘α∙ is implied by the fact that, if U U UUU is tidy for g g ggg, then U U UUU is also tidy for all h U g U h ∈ U g U h in UgUh \in U g Uh∈UgU and s ( h ) = s ( g ) s ( h ) = s ( g ) s(h)=s(g)s(h)=s(g)s(h)=s(g), [85, THEOREM 3].
Questions about the structure of tdlc groups may be answered with scale and tidy subgroup techniques. K. H. Hofmann and A. Mukherjea conjectured in [45] that all locally compact groups are "neat"-a property involving the conjugation action by a single element g g ggg. They used approximation by Lie groups to reduce to the totally disconnected case, and subgroups tidy for g g ggg are used in [47] to show that all groups are neat. Answering another question of K.H. Hofmann, the set per ( G ) per ⁡ ( G ) per(G)\operatorname{per}(G)per⁡(G), comprising those elements of G G GGG such that the closure of g ⟨ g ⟩ (:g:)\langle g\rangle⟨g⟩ is compact, is shown in [84] to be closed by appealing to the properties of the scale given in Theorem 4.2.
The scale and the subgroup U + U + U_(+)U_{+}U+associated with it in Theorem 4.1 are given a concrete representation in [9]. Put U + + = n 0 α n ( U + ) U + + = ⋃ n ≥ 0   α n U + U_(++)=uuu_(n >= 0)alpha^(n)(U_(+))U_{++}=\bigcup_{n \geq 0} \alpha^{n}\left(U_{+}\right)U++=⋃n≥0αn(U+). Then U + + U + + U_(++)U_{++}U++is closed if U U UUU is tidy and U + + α U + + ⋊ ⟨ α ⟩ U_(++)><|(:alpha:)U_{++} \rtimes\langle\alpha\rangleU++⋊⟨α⟩ acts on a regular tree with valency s ( α ) + 1 s ( α ) + 1 s(alpha)+1s(\alpha)+1s(α)+1 : the image of U + + α U + + ⋊ ⟨ α ⟩ U_(++)><|(:alpha:)U_{++} \rtimes\langle\alpha\rangleU++⋊⟨α⟩ is a closed subgroup of the isometry group of the tree; is transitive on vertices; and fixes an end of the tree. The resulting isometry groups of trees correspond to the self-replicating groups studied in [58]. Moreover, the semidirect product U + + α U + + ⋊ ⟨ α ⟩ U_(++)><|(:alpha:)U_{++} \rtimes\langle\alpha\rangleU++⋊⟨α⟩ also belongs to the family of focal hyperbolic groups studied in [18].

4.1. Contraction and other groups

Subgroups of G G GGG defined in terms of the action of α α alpha\alphaα are related to the scale and tidy subgroups. It is convenient to confine the statements to automorphisms here. Extensions to endomorphisms may be found in [ 16 , 89 ] [ 16 , 89 ] [16,89][16,89][16,89].
The contraction subgroup for α Aut ( G ) α ∈ Aut ⁡ ( G ) alpha in Aut(G)\alpha \in \operatorname{Aut}(G)α∈Aut⁡(G) is
con ( α ) = { x G α n ( x ) 1 as n } con ⁡ ( α ) = x ∈ G ∣ α n ( x ) → 1  as  n → ∞ con(alpha)={x in G∣alpha^(n)(x)rarr1" as "n rarr oo}\operatorname{con}(\alpha)=\left\{x \in G \mid \alpha^{n}(x) \rightarrow 1 \text { as } n \rightarrow \infty\right\}con⁡(α)={x∈G∣αn(x)→1 as n→∞}
The next result, from [ 9 , 46 ] [ 9 , 46 ] [9,46][9,46][9,46], relates contraction subgroups to the scale.
Theorem 4.3. Let α Aut ( G ) α ∈ Aut ⁡ ( G ) alpha in Aut(G)\alpha \in \operatorname{Aut}(G)α∈Aut⁡(G). Then { U U â‹‚ U − − ∣ U nnn{U_(--)∣U:}\bigcap\left\{U_{--} \mid U\right.â‹‚{U−−∣U is tidy for α } α {: alpha}\left.\alpha\right\}α} is equal to con ( α ) ¯ con ⁡ ( α ) ¯ bar(con(alpha))\overline{\operatorname{con}(\alpha)}con⁡(α)¯, and s ( α 1 ) s α − 1 s(alpha^(-1))s\left(\alpha^{-1}\right)s(α−1) is equal to the scale of the restriction of α 1 α − 1 alpha^(-1)\alpha^{-1}α−1 to con ( α ) ¯ con ⁡ ( α ) ¯ bar(con(alpha))\overline{\operatorname{con}(\alpha)}con⁡(α)¯. Hence s ( α 1 ) > 1 s α − 1 > 1 s(alpha^(-1)) > 1s\left(\alpha^{-1}\right)>1s(α−1)>1 if and only if con ( α ) ¯ con ⁡ ( α ) ¯ bar(con(alpha))\overline{\operatorname{con}(\alpha)}con⁡(α)¯ is not compact.
If G G GGG is a p p ppp-adic Lie group, then con ( α ) con ⁡ ( α ) con(alpha)\operatorname{con}(\alpha)con⁡(α) is closed for every α α alpha\alphaα, [80], but that is not the case if, for example, G G GGG is the isometry group of a regular tree, or a certain type of complete Kac-Moody group [7], or if L E ( G ) { 0 , } L E ( G ) ≠ { 0 , ∞ } LE(G)!={0,oo}\mathscr{L} \mathscr{E}(G) \neq\{0, \infty\}LE(G)≠{0,∞} [28]. The closedness of con ( α ) ( α ) (alpha)(\alpha)(α) is equivalent, by [9, THEOREM 3.32], to the triviality of the n u b n u b nubn u bnub subgroup,
nub ( α ) = { U U tidy for α } nub ⁡ ( α ) = â‹‚ { U ∣ U  tidy for  α } nub(alpha)=nnn{U∣U" tidy for "alpha}\operatorname{nub}(\alpha)=\bigcap\{U \mid U \text { tidy for } \alpha\}nub⁡(α)=â‹‚{U∣U tidy for α}
The nub for α α alpha\alphaα is compact and is the largest α α alpha\alphaα-stable subgroup of G G GGG on which α α alpha\alphaα acts ergodically, which sharpens the theorem of N. Aoki in [2] that a totally disconnected locally compact group with an ergodic automorphism must be compact. P. Halmos had asked in [44] whether that was so for all locally compact groups. See [48,90] for the connected case, and also [60].
The structure of closed contraction subgroups con ( α ) ( α ) (alpha)(\alpha)(α) is described precisely in [40] If con ( α ) con ⁡ ( α ) con(alpha)\operatorname{con}(\alpha)con⁡(α) is closed, there is a composition series
{ 1 } = G 0 G n = con ( α ) { 1 } = G 0 â—ƒ ⋯ â—ƒ G n = con ⁡ ( α ) {1}=G_(0)â—ƒcdotsâ—ƒG_(n)=con(alpha)\{1\}=G_{0} \triangleleft \cdots \triangleleft G_{n}=\operatorname{con}(\alpha){1}=G0◃⋯◃Gn=con⁡(α)
of α α alpha\alphaα-stable closed subgroups of con ( α ) con ⁡ ( α ) con(alpha)\operatorname{con}(\alpha)con⁡(α) such that the factors G i + 1 / G i G i + 1 / G i G_(i+1)//G_(i)G_{i+1} / G_{i}Gi+1/Gi have no proper, nontrivial α α alpha\alphaα-stable closed subgroups. The factors appearing in any such series are unique up to permutation and isomorphism, and their isomorphism types come from a countable list: each torsion factor being a restricted product i Z ( G i , U i ) ⨁ i ∈ Z   G i , U i bigoplus_(i inZ)(G_(i),U_(i))\bigoplus_{i \in \mathbf{Z}}\left(G_{i}, U_{i}\right)⨁i∈Z(Gi,Ui) with G i = F G i = F G_(i)=FG_{i}=FGi=F, a finite simple group, and U i = F U i = F U_(i)=FU_{i}=FUi=F if i 0 i ≥ 0 i >= 0i \geq 0i≥0 and trivial if i < 0 i < 0 i < 0i<0i<0, and the automorphism the shift; and each divisible factor being a p p ppp-adic vector group and the automorphism a linear transformation. Moreover, con ( α ) ( α ) (alpha)(\alpha)(α) is the direct product T × D T × D T xx DT \times DT×D with T T TTT a torsion and D D DDD a divisible α α alpha\alphaα-stable subgroup. The divisible subgroup D D DDD is a direct product D p 1 × × D p r D p 1 × ⋯ × D p r D_(p_(1))xx cdots xxD_(p_(r))D_{p_{1}} \times \cdots \times D_{p_{r}}Dp1×⋯×Dpr with D p i D p i D_(p_(i))D_{p_{i}}Dpi a nilpotent p p ppp-adic Lie group for each p i p i p_(i)p_{i}pi. The torsion group T T TTT may include nonabelian irreducible factors but, should it happen to be locally pro- p p ppp, then it is nilpotent too, see [42]. The number of nonisomorphic locally pro- p p ppp closed contraction groups is uncountable [41].
Contraction groups correspond to unipotent subgroups of algebraic groups and, following [75], the Tits core, G G † G^(†)G^{\dagger}G†, of the tdlc group G G GGG is defined to be the subgroup generated by all closures of contraction groups. It is shown in [26] that, if G G GGG is topologically simple, then G G † G^(†)G^{\dagger}G† is either trivial or is abstractly simple and dense in G G GGG.
The correspondence with algebraic groups is pursued in [9], where the parabolic subgroup for α Aut ( G ) α ∈ Aut ⁡ ( G ) alpha in Aut(G)\alpha \in \operatorname{Aut}(G)α∈Aut⁡(G) is defined to be
par ( α ) = { x G { α n ( x ) } n 0 has compact closure } par ⁡ ( α ) = x ∈ G ∣ α n ( x ) n ≥ 0  has compact closure  par(alpha)={x in G∣{alpha^(n)(x)}_(n >= 0)" has compact closure "}\operatorname{par}(\alpha)=\left\{x \in G \mid\left\{\alpha^{n}(x)\right\}_{n \geq 0} \text { has compact closure }\right\}par⁡(α)={x∈G∣{αn(x)}n≥0 has compact closure }
and the Levi factor to be lev ( α ) = par ( α ) par ( α 1 ) lev ⁡ ( α ) = par ⁡ ( α ) ∩ par ⁡ α − 1 lev(alpha)=par(alpha)nn par(alpha^(-1))\operatorname{lev}(\alpha)=\operatorname{par}(\alpha) \cap \operatorname{par}\left(\alpha^{-1}\right)lev⁡(α)=par⁡(α)∩par⁡(α−1). Then par ( α ) par ⁡ ( α ) par(alpha)\operatorname{par}(\alpha)par⁡(α), and hence lev ( α ) lev ⁡ ( α ) lev(alpha)\operatorname{lev}(\alpha)lev⁡(α), is closed in G G GGG, [85, PRoposition 3]. It may be verified that con ( α ) par ( α ) con ⁡ ( α ) â—ƒ par ⁡ ( α ) con(alpha)â—ƒpar(alpha)\operatorname{con}(\alpha) \triangleleft \operatorname{par}(\alpha)con⁡(α)â—ƒpar⁡(α) and shown, see [9], that par ( α ) = lev ( α ) con ( α ) par ⁡ ( α ) = lev ⁡ ( α ) con ⁡ ( α ) par(alpha)=lev(alpha)con(alpha)\operatorname{par}(\alpha)=\operatorname{lev}(\alpha) \operatorname{con}(\alpha)par⁡(α)=lev⁡(α)con⁡(α).

4.2. Flat groups of automorphisms

A group, H H H\mathscr{H}H, of automorphisms of G G GGG is flat if there is a compact open subgroup, U G U ≤ G U <= GU \leq GU≤G, that is tidy for every α H α ∈ H alpha inH\alpha \in \mathscr{H}α∈H. The stabilizer of U U UUU in H H H\mathscr{H}H is called the uniscalar subgroup and denoted H u H u H_(u)\mathscr{H}_{u}Hu. The factoring of subgroups tidy for a single automorphism in Theorem 4.1 extends to flat groups as follows.
Theorem 4.4 ([87]). Let H H H\mathscr{H}H be a finitely generated flat group of automorphisms of G G GGG and suppose that U U UUU is tidy for H H H\mathscr{H}H. Then H u H H u ◃ H H_(u)◃H\mathscr{H}_{u} \triangleleft \mathscr{H}Hu◃H and there is r 0 r ≥ 0 r >= 0r \geq 0r≥0 such that
H / H u Z r H / H u ≅ Z r H//H_(u)~=Z^(r)\mathscr{H} / \mathscr{H}_{u} \cong \mathbb{Z}^{r}H/Hu≅Zr
  • There are q 0 q ≥ 0 q >= 0q \geq 0q≥0 and closed groups U j U , j { 0 , 1 , , q } U j ≤ U , j ∈ { 0 , 1 , … , q } U_(j) <= U,j in{0,1,dots,q}U_{j} \leq U, j \in\{0,1, \ldots, q\}Uj≤U,j∈{0,1,…,q} such that α ( U 0 ) = U 0 ; α ( U j ) α U 0 = U 0 ; α U j alpha(U_(0))=U_(0);alpha(U_(j))\alpha\left(U_{0}\right)=U_{0} ; \alpha\left(U_{j}\right)α(U0)=U0;α(Uj) is either a subgroup or supergroup of U j U j U_(j)U_{j}Uj for every j { 1 , , q } j ∈ { 1 , … , q } j in{1,dots,q}j \in\{1, \ldots, q\}j∈{1,…,q}; and U = U 0 U 1 U q U = U 0 U 1 ⋯ U q U=U_(0)U_(1)cdotsU_(q)U=U_{0} U_{1} \cdots U_{q}U=U0U1⋯Uq.
  • U ~ j := α H α ( U j ) U ~ j := ⋃ α ∈ H   α U j tilde(U)_(j):=uuu_(alpha inH)alpha(U_(j))\tilde{U}_{j}:=\bigcup_{\alpha \in \mathscr{H}} \alpha\left(U_{j}\right)U~j:=⋃α∈Hα(Uj) is a closed subgroup of G for each j { 1 , , q } j ∈ { 1 , … , q } j in{1,dots,q}j \in\{1, \ldots, q\}j∈{1,…,q}.
  • There are, for each j { 1 , , q } j ∈ { 1 , … , q } j in{1,dots,q}j \in\{1, \ldots, q\}j∈{1,…,q}, an integer s j > 1 s j > 1 s_(j) > 1s_{j}>1sj>1 and a surjective homomorphism ρ j : H ( Z , + ) ρ j : H → ( Z , + ) rho_(j):Hrarr(Z,+)\rho_{j}: \mathscr{H} \rightarrow(\mathbb{Z},+)ρj:H→(Z,+) such that Δ ( α | U ~ j ) = s j ρ j ( α ) Δ α U ~ j = s j ρ j ( α ) Delta( alpha|_( tilde(U)_(j)))=s_(j)^(rho_(j)(alpha))\Delta\left(\left.\alpha\right|_{\tilde{U}_{j}}\right)=s_{j}^{\rho_{j}(\alpha)}Δ(α|U~j)=sjρj(α).
  • The integers r r rrr and q q qqq, and integers s j s j s_(j)s_{j}sj and homomorphisms ρ j ρ j rho_(j)\rho_{j}ρj for each j { 1 , , q } j ∈ { 1 , … , q } j in{1,dots,q}j \in\{1, \ldots, q\}j∈{1,…,q}, are independent of the subgroup U U UUU tidy for H H H\mathscr{H}H.
The number r r rrr in Theorem 4.4 is the flat rank of H H H\mathscr{H}H. The singly-generated group α ⟨ α ⟩ (:alpha:)\langle\alpha\rangle⟨α⟩ has flat rank equal to 0 if α α alpha\alphaα is uniscalar and 1 if not. Flat groups of automorphisms with rank at least 1 correspond to Cartan subgroups in Lie groups over local fields and may be interpreted geometrically in terms of apartments in isometry groups of buildings [8].
More generally, flatness of groups of automorphisms may be shown by the following converse to the fact that flat groups are abelian modulo the stablizer of tidy subgroups.
Theorem 4.5 ([87][71]). Every finitely generated nilpotent subgroup of Aut ( G ) Aut ⁡ ( G ) Aut(G)\operatorname{Aut}(G)Aut⁡(G) is flat, and every polycyclic subgroup is virtually flat.
Flatness is used-in combination with bounded generation of arithmetic groups [57,74], the fact that almost normal subgroups are close to normal [11], and the Margulis normal subgroup theorem [53]-to prove the Margulis-Zimmer conjecture in the special case of Chevalley groups in [71] and show that there are no commensurated subgroups of arithmetic subgroups other than the natural ones.

5. FUTURE DIRECTIONS

The contributions to the structure theory of tdlc groups surveyed in this article highlight that, for a general tdlc group G G GGG, as soon as the topology is nondiscrete, its interaction with the group structure yields significant algebraic constraints. As mentioned in the introduction, we view the dynamics of the conjugation action as a unifying theme of our considerations. The results we have surveyed reveal that those dynamics tend to be richer
than one might expect. This is especially the case among tdlc groups that are nonelementary. We hope that further advances will shed more light on this paradigm in the future.
Concerning decomposition theory, it is an important open problem to clarify what distinguishes elementary and nonelementary tdlc groups. A key question asks whether every nonelementary tdlcsc group G G GGG contains a closed subgroup H H HHH admitting a quotient in S td S td  S_("td ")\mathscr{S}_{\text {td }}Std . Concerning simple groups, our results yield a dichotomy, depending on whether the centralizer lattice is trivial or not. The huge majority of known examples of groups in S td S td  S_("td ")\mathscr{S}_{\text {td }}Std  (listed in [28, APPENDIX A]) have a nontrivial centralizer lattice, the most notable exceptions being the simple algebraic groups over local fields. Finding new groups in S t d S t d S_(td)\mathscr{S}_{\mathrm{td}}Std with a trivial centralizer lattice would be a decisive step forward. A fundamental source of examples of tdlc groups is provided by Galois groups of transcendental field extensions with finite transcendence degree (see [66, TH. 2.9], highlighting the occurrence of topologically simple groups), but this territory remains largely unexplored from the viewpoint of structure theory of tdlc groups. Concerning scale methods, the structure of tdlc groups all of whose elements are uniscalar (i.e., have scale 1) is still mysterious. In particular, we do not know whether every such group is elementary. This is equivalent to asking whether a tdlc group, all of whose closed subgroups are unimodular, is necessarily elementary. A positive answer would provide a formal incarnation to the claim that the dynamics of the conjugation action is nontrivial for all nonelementary tdlc groups. We refer to [24] for a more extensive list of specific problems.
We believe that a good measurement of the maturity of a mathematical theory is provided by its ability to solve problems arising on the outside of the theory. For the structure theory of tdlc groups, the Margulis-Zimmer conjecture appears as a natural target. As mentioned in Section 4, partial results in the nonuniform case, relying on scale methods on tdlc groups, have already been obtained in [71].
Another source of external problems is provided by abstract harmonic analysis. As mentioned in the introduction, the emergence of locally compact groups as an independent subject of study coincides with the foundation of abstract harmonic analysis. However, fundamental problems clarifying the links between the algebraic structure of a locally compact group and the properties of its unitary representations remain open. The class of amenable locally compact groups is defined by a representation theoretic property (indeed, a locally compact group is amenable if and only if every unitary representation is weakly contained in the regular), but purely algebraic characterizations of amenable groups are still missing. In particular, the following nondiscrete version of Day's problem is open and intriguing: Is every amenable second countable tdlc group elementary (in the sense of Section 2)? The unitary representation theory also reveals a fundamental dichotomy between locally compact groups of type I I III (roughly speaking, those for which the problem of classifying the irreducible unitary representations up to equivalence is tractable) and the others (see [ 10 , 35 , 52 ] [ 10 , 35 , 52 ] [10,35,52][10,35,52][10,35,52] ). Algebraic characterizations of type I groups are also desirable. In particular, we underline the following question: Does every second countable locally compact group of type I contain a cocompact amenable subgroup? For a more detailed discussion of that problem and related results, we refer to [20].

ACKNOWLEDGMENTS

It is a pleasure to take this opportunity to heartily thank our collaborators, past and present, for their ideas and companionship. We are grateful to Adrien Le Boudec and Colin Reid for their comments on a preliminary version of this article.

FUNDING

This work was written while P.-E. Caprace was a Senior Research Associate of the Belgian F.R.S.-FNRS and G. A. Willis was Australian Research Council Laureate Professor under the grant FL170100032. The support of these bodies is gratefully acknowledged.

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PIERRE-EMMANUEL CAPRACE

UCLouvain - IRMP, 1348 Louvain-la-Neuve, Belgium,

GEORGE A. WILLIS

University of Newcastle, Callaghan 2308, Australia, george.willis @ newcastle.edu.au

NEENA GUPTA

ABSTRACT

In this article, we shall discuss the solution to the Zarsiki Cancellation Problem in positive characteristic, various approaches taken so far towards the possible solution in characteristic zero, and several other questions related to this problem.

MATHEMATICS SUBJECT CLASSIFICATION 2020

Primary 14R10; Secondary 14R20,14R25,13B25,13F20,13N15

KEYWORDS

Polynomial ring, cancellation Problem, embedding problem, affine fibration problem, locally nilpotent derivations

1. INTRODUCTION

"Polynomials and power series
May they forever rule the world."
Right from the beginning of the 19th century, mathematicians have been involved in studying polynomial rings (over C C C\mathbb{C}C and over R R R\mathbb{R}R ). Some of the early breakthroughs on polynomial rings have led to the foundation of Commutative Algebra. One such result is the Hilbert Basis Theorem, a landmark result on the finite generation of ideals, which solved a central problem on invariant theory. This was followed by the Hilbert Nullstellensatz which connects affine varieties (zero locus of a set of polynomials) with rings of regular functions on varieties and thus enables one to make use of the algebraic machinery of commutative algebra to study geometric properties of varieties.
Affine Algebraic Geometry deals with the study of affine spaces (and certain closed subspaces), equivalently, polynomial rings (and certain quotients). There are many fundamental problems on polynomial rings which can be formulated in an elementary mathematical language but whose solutions remain elusive. Any significant progress requires the development of new and powerful methods and their ingenious applications.
One of the most challenging problems in Affine Algebraic Geometry is the Zariski Cancelation Problem (ZCP) on polynomial rings (Question 1' below). In this article, we shall discuss the solution to the ZCP in positive characteristic, various approaches taken so far towards the possible solution in characteristic zero, and several other questions related to this problem. For a survey on problems in Affine Algebraic Geometry, one may look at [ 42 , 62 , 69 ] [ 42 , 62 , 69 ] [42,62,69][42,62,69][42,62,69].
Throughout the article, all rings will be assumed to be commutative with unity and k k kkk will denote a field. For a ring R , R R , R ∗ R,R^(**)R, R^{*}R,R∗ will denote the group of units of R R RRR. We shall use the notation R [ n ] R [ n ] R^([n])R^{[n]}R[n] for a polynomial ring in n n nnn variables over a commutative ring R R RRR. Thus, E = R [ n ] E = R [ n ] E=R^([n])E=R^{[n]}E=R[n] will mean that E = R [ t 1 , , t n ] E = R t 1 , … , t n E=R[t_(1),dots,t_(n)]E=R\left[t_{1}, \ldots, t_{n}\right]E=R[t1,…,tn] for some elements t 1 , , t n t 1 , … , t n t_(1),dots,t_(n)t_{1}, \ldots, t_{n}t1,…,tn in E E EEE which are algebraically independent over R R RRR. Unless otherwise stated, capital letters like X 1 , X 2 , , X n , Y 1 , , Y m , X , Y , Z , T X 1 , X 2 , … , X n , Y 1 , … , Y m , X , Y , Z , T X_(1),X_(2),dots,X_(n),Y_(1),dots,Y_(m),X,Y,Z,TX_{1}, X_{2}, \ldots, X_{n}, Y_{1}, \ldots, Y_{m}, X, Y, Z, TX1,X2,…,Xn,Y1,…,Ym,X,Y,Z,T will be used as variables of polynomial rings.

2. CANCELLATION PROBLEM

Let A A AAA be an affine (finitely generated) algebra over a field k k kkk. The k k kkk-algebra A A A\mathrm{A}A is said to be cancellative (over k k kkk ) if, for any k k kkk-algebra B , A [ X ] k B [ X ] B , A [ X ] ≅ k B [ X ] B,A[X]~=_(k)B[X]B, A[X] \cong_{k} B[X]B,A[X]≅kB[X] implies that A k B A ≅ k B A~=_(k)BA \cong_{k} BA≅kB. A natural question in this regard is: which affine domains are cancellative? More precisely:
Question 1. Let A A AAA be an affine algebra over a field k k kkk. Suppose that B B BBB is a k k kkk-algebra such that the polynomial rings A [ X ] A [ X ] A[X]A[X]A[X] and B [ X ] B [ X ] B[X]B[X]B[X] are isomorphic as k k kkk-algebras. Does it follow that A k B A ≅ k B A~=_(k)BA \cong_{k} BA≅kB ? In other words, is the k k kkk-algebra A A AAA cancellative?
A special case of Question 1, famously known as the Zariski Cancellation Problem, asks whether affine spaces are cancellative, i.e., whether any polynomial ring in n n nnn variables over a field k k kkk is cancellative. More precisely:
Question 1'. Suppose that B B BBB is an affine k k kkk algebra satisfying B [ X ] k k [ X 1 , , X n + 1 ] B [ X ] ≅ k k X 1 , … , X n + 1 B[X]~=_(k)k[X_(1),dots,X_(n+1)]B[X] \cong_{k} k\left[X_{1}, \ldots, X_{n+1}\right]B[X]≅kk[X1,…,Xn+1] for some positive integer n n nnn. Does it follow that B k k [ X 1 , , X n ] B ≅ k k X 1 , … , X n B~=_(k)k[X_(1),dots,X_(n)]B \cong_{k} k\left[X_{1}, \ldots, X_{n}\right]B≅kk[X1,…,Xn] ? In other words, is the polynomial ring k [ X 1 , , X n ] k X 1 , … , X n k[X_(1),dots,X_(n)]k\left[X_{1}, \ldots, X_{n}\right]k[X1,…,Xn] cancellative?
Abhyankar, Eakin, and Heinzer have shown that any domain A A AAA of transcendence degree one over any field k k kkk is cancellative [3]. In fact, they showed that, for any UFD R R RRR, the polynomial ring R [ X ] R [ X ] R[X]R[X]R[X] is cancellative over R R RRR. This was further generalized by Hamann to a ring R R RRR which either contains Q Q Q\mathbb{Q}Q or is a seminormal domain [52].
In 1972, Hochster demonstrated the first counterexample to Question 1 [53]. His example, a four-dimensional ring over the field of real numbers R R R\mathbb{R}R, is based on the fact that the projective module defined by the tangent bundle over the real sphere with coordinate ring S = R [ X , Y , Z ] / ( X 2 + Y 2 + Z 2 1 ) S = R [ X , Y , Z ] / X 2 + Y 2 + Z 2 − 1 S=R[X,Y,Z]//(X^(2)+Y^(2)+Z^(2)-1)S=\mathbb{R}[X, Y, Z] /\left(X^{2}+Y^{2}+Z^{2}-1\right)S=R[X,Y,Z]/(X2+Y2+Z2−1) is stably free but not a free S S SSS-module.
One of the major breakthroughs in 1970s was the establishment of an affirmative answer to Question 1' for the case n = 2 n = 2 n=2n=2n=2. This was proved over a field of characteristic zero by Fujita, Miyanishi, and Sugie [43,70] and over perfect fields of arbitrary characteristic by Russell [74]. Later, it has been shown that even the hypothesis of perfect field can be dropped [20]. A simplified proof of the cancellation property of k [ X , Y ] k [ X , Y ] k[X,Y]k[X, Y]k[X,Y] for an algebraically closed field k k kkk is given by Crachiola and Makar-Limanov in [22].
Around 1989, Danielewski [26] constructed explicit two-dimensional affine domains over the field of complex numbers C C C\mathbb{C}C which are not cancellative over C C C\mathbb{C}C. New examples of noncancellative varieties over any field k k kkk have been studied in [ 9 , 32 , 49 ] [ 9 , 32 , 49 ] [9,32,49][9,32,49][9,32,49]. This addresses the Cancellation Problem, as formulated in Question 1, for all dimensions.
In [45] and [47], the author settled the Zariski Cancellation Problem (Question 1') completely for affine spaces in positive characteristic. She has first shown in [45] that a certain threefold constructed by Asanuma is a counterexample to the ZCP in positive characteristic for the affine three space. Later in [46], she studied a general threefold of the form x m y = F ( x , z , t ) x m y = F ( x , z , t ) x^(m)y=F(x,z,t)x^{m} y=F(x, z, t)xmy=F(x,z,t), which includes the Asanuma threefold as well as the famous Russell cubic defined below. A major theorem of [46] is stated as Theorem 5.4 of this article. In [47], using a modification of the theory developed in [46], she constructed a family of examples which are counterexamples to the ZCP in positive characteristic in all dimensions greater than 2 . The ZCP is still a challenging problem in characteristic zero. A few candidate counterexamples are discussed below.
The Russell cubic. Let A = C [ X , Y , Z , T ] / ( X 2 Y + X + Z 2 + T 3 ) , V = Spec A A = C [ X , Y , Z , T ] / X 2 Y + X + Z 2 + T 3 , V = Spec ⁡ A A=C[X,Y,Z,T]//(X^(2)Y+X+Z^(2)+T^(3)),V=Spec AA=\mathbb{C}[X, Y, Z, T] /\left(X^{2} Y+X+Z^{2}+T^{3}\right), V=\operatorname{Spec} AA=C[X,Y,Z,T]/(X2Y+X+Z2+T3),V=Spec⁡A and let x x xxx denote the image of X X XXX in A A AAA. The ring A A AAA, known as the Russell cubic, is one of the simplest examples of the Koras-Russell threefolds, a family of threefolds which arose in the context of the problem of determining whether there exist nonlinearizable C C ∗ C^(**)\mathbb{C}^{*}C∗-actions on C 3 C 3 C^(3)\mathbb{C}^{3}C3. It was an exciting open problem for some time whether A C [ 3 ] A ≅ C [ 3 ] A~=C^([3])A \cong \mathbb{C}^{[3]}A≅C[3]. It was first observed that the ring
A A AAA (respectively the variety V V VVV ) has several properties in common with C [ 3 ] C [ 3 ] C^([3])\mathbb{C}^{[3]}C[3] (respectively C 3 C 3 C^(3)\mathbb{C}^{3}C3 ), for instance,
(i) A A AAA is a regular UFD.
(ii) There exists an injective C C C\mathbb{C}C-algebra homomorphism from A A AAA to C [ 3 ] C [ 3 ] C^([3])\mathbb{C}^{[3]}C[3]. Note that C [ 3 ] A C [ 3 ] ↪ A C^([3])↪A\mathbb{C}^{[3]} \hookrightarrow AC[3]↪A.
(iii) The variety V V VVV is homeomorphic (in fact, diffeomorphic) to R 6 R 6 R^(6)\mathbb{R}^{6}R6.
(iv) V V quad V\quad VV has logarithmic Kodaira dimension − ∞ -oo-\infty−∞.
These properties appeared to provide evidence in favor of the surmise that A C [ 3 ] A ≅ C [ 3 ] A~=C^([3])A \cong \mathbb{C}^{[3]}A≅C[3]. The establishment of an isomorphism between A A AAA and C [ 3 ] C [ 3 ] C^([3])\mathbb{C}^{[3]}C[3] would have led to counterexamples to the "Linearization Conjecture" on C 3 C 3 C^(3)\mathbb{C}^{3}C3 (stated in [58]) and the Abhyankar-Sathaye Conjecture for n = 3 n = 3 n=3n=3n=3 (stated in Section 5 of the present article). Indeed, if A A AAA were isomorphic to C [ 3 ] C [ 3 ] C^([3])\mathbb{C}^{[3]}C[3], as was then suspected, it would have shown the existence of nonlinearizable C C ∗ C^(**)\mathbb{C}^{*}C∗-actions on C 3 C 3 C^(3)\mathbb{C}^{3}C3. Moreover, note that
(v) A / ( x λ ) = C [ 2 ] A / ( x − λ ) = C [ 2 ] A//(x-lambda)=C^([2])A /(x-\lambda)=\mathbb{C}^{[2]}A/(x−λ)=C[2] for every λ C λ ∈ C ∗ lambda inC^(**)\lambda \in \mathbb{C}^{*}λ∈C∗.
(vi) A / ( x ) C [ 2 ] A / ( x ) ≠ C [ 2 ] A//(x)!=C^([2])A /(x) \neq \mathbb{C}^{[2]}A/(x)≠C[2].
Therefore, if A A AAA were isomorphic to C [ 3 ] C [ 3 ] C^([3])\mathbb{C}^{[3]}C[3], then property (vi) would show that x λ x − λ x-lambdax-\lambdax−λ cannot be a coordinate in A A AAA for any λ λ lambda\lambdaλ and then, by property (v), it would have yielded a counterexample to the Abhyankar-Sathaye Conjecture for n = 3 n = 3 n=3n=3n=3.
However, Makar-Limanov proved [65] that A C [ 3 ] A ≠ C [ 3 ] A!=C^([3])A \neq \mathbb{C}^{[3]}A≠C[3]; for this result, he introduced a new invariant which distinguished between A A AAA and C [ 3 ] C [ 3 ] C^([3])\mathbb{C}^{[3]}C[3]. This invariant, which he had named AK-invariant, is now named Makar-Limanov invariant and is denoted by ML. It is defined in Section 3. Makar-Limanov proved that
(vii) ML ( A ) = C [ x ] ML ⁡ ( A ) = C [ x ] ML(A)=C[x]\operatorname{ML}(A)=\mathbb{C}[x]ML⁡(A)=C[x] (Makar-Limanov [65]).
However, the Makar-Limanov invariant of C [ n ] C [ n ] C^([n])\mathbb{C}^{[n]}C[n] is C C C\mathbb{C}C for any integer n 1 n ≥ 1 n >= 1n \geq 1n≥1. Thus A C [ 3 ] A ⊈ C [ 3 ] A⊈C^([3])A \nsubseteq \mathbb{C}^{[3]}A⊈C[3]. Subsequently, other Koras-Russell threefolds were shown to be not isomorphic to the polynomial ring. Eventually, Kaliman-Koras-Makar-Limanov-Russell proved that every C C ∗ C^(**)\mathbb{C}^{*}C∗-action on C 3 C 3 C^(3)\mathbb{C}^{3}C3 is linearizable (cf. [55]).
Now for ZCP in characteristic zero, a crucial question, still open, is whether A [ 1 ] = C [ 4 ] A [ 1 ] = C [ 4 ] A^([1])=C^([4])A^{[1]}=\mathbb{C}^{[4]}A[1]=C[4]. Because if A [ 1 ] = C [ 4 ] A [ 1 ] = C [ 4 ] A^([1])=C^([4])A^{[1]}=\mathbb{C}^{[4]}A[1]=C[4], then A A AAA would be a counterexample to the ZCP in characteristic zero for n = 3 n = 3 n=3n=3n=3. In this context, the following results have been proved:
(viii) ML ( A [ 1 ] ) = C ML ⁡ A [ 1 ] = C ML(A^([1]))=C\operatorname{ML}\left(A^{[1]}\right)=\mathbb{C}ML⁡(A[1])=C (Dubouloz [30]).
(ix) V V quad V\quad VV is A 1 A 1 A^(1)\mathbb{A}^{1}A1-contractible (Dubouloz-Fasel [31], also see [33,54]).
Note that A [ 1 ] = C [ 4 ] A [ 1 ] = C [ 4 ] A^([1])=C^([4])A^{[1]}=\mathbb{C}^{[4]}A[1]=C[4] would imply that ML ( A [ 1 ] ) = C ML ⁡ A [ 1 ] = C ML(A^([1]))=C\operatorname{ML}\left(A^{[1]}\right)=\mathbb{C}ML⁡(A[1])=C and Dubouloz's result (viii) shows that the latter indeed holds. On the other hand, Asok had suggested a program for showing that the variety V V VVV is not A 1 A 1 A^(1)\mathbb{A}^{1}A1-contractible and hence A A A\mathrm{A}A is not a stably polynomial
ring (see [54]). However, Hoyois, Krishna, and Østvær have proved [54] that a step in his program does not hold for V V VVV. They had further shown that V V VVV is stably A 1 A 1 A^(1)\mathbb{A}^{1}A1-contractible. In a remarkable paper [31], Dubouloz and Fasel have established that V V VVV is in fact A 1 A 1 A^(1)\mathbb{A}^{1}A1-contractible, which seems to provide further evidence in favor of A [ 1 ] = C [ 4 ] A [ 1 ] = C [ 4 ] A^([1])=C^([4])A^{[1]}=\mathbb{C}^{[4]}A[1]=C[4]. The variety V V VVV is in fact the first example of an A 1 A 1 A^(1)\mathbb{A}^{1}A1-contractible threefold which is not algebraically isomorphic to C 3 C 3 C^(3)\mathbb{C}^{3}C3.
Nonrectifiable epimorphisms and Asanuma's rings. Let m n m ≤ n m <= nm \leq nm≤n be two integers. A k k kkk-algebra epimorphism ϕ : k [ X 1 , , X n ] k [ Y 1 , , Y m ] Ï• : k X 1 , … , X n → k Y 1 , … , Y m phi:k[X_(1),dots,X_(n)]rarr k[Y_(1),dots,Y_(m)]\phi: k\left[X_{1}, \ldots, X_{n}\right] \rightarrow k\left[Y_{1}, \ldots, Y_{m}\right]Ï•:k[X1,…,Xn]→k[Y1,…,Ym] is said to be rectifiable if there exists a k k kkk-algebra automorphism ψ ψ psi\psiψ of k [ X 1 , , X n ] k X 1 , … , X n k[X_(1),dots,X_(n)]k\left[X_{1}, \ldots, X_{n}\right]k[X1,…,Xn] such that ϕ ψ ( X i ) = Y i Ï• ∘ ψ X i = Y i phi_(@)psi(X_(i))=Y_(i)\phi_{\circ} \psi\left(X_{i}\right)=Y_{i}ϕ∘ψ(Xi)=Yi for 1 i m 1 ≤ i ≤ m 1 <= i <= m1 \leq i \leq m1≤i≤m and ϕ ψ ( X j ) = 0 Ï• ∘ ψ X j = 0 phi_(@)psi(X_(j))=0\phi_{\circ} \psi\left(X_{j}\right)=0ϕ∘ψ(Xj)=0 for m + 1 j n m + 1 ≤ j ≤ n m+1 <= j <= nm+1 \leq j \leq nm+1≤j≤n. Equivalently, over an algebraically closed field k k kkk, a k k kkk embedding Φ : A k m A k n Φ : A k m ↪ A k n Phi:A_(k)^(m)↪A_(k)^(n)\Phi: \mathbb{A}_{k}^{m} \hookrightarrow \mathbb{A}_{k}^{n}Φ:Akm↪Akn is said to be rectifiable if there exists an automorphism Ψ Î¨ Psi\PsiΨ of A k n A k n A_(k)^(n)\mathbb{A}_{k}^{n}Akn such that Ψ Φ Î¨ ∘ Φ Psi_(@)Phi\Psi_{\circ} \PhiΨ∘Φ is the canonical embedding mapping ( y 1 , , y m ) ( y 1 , , y m , 0 , , 0 ) y 1 , … , y m → y 1 , … , y m , 0 , … , 0 (y_(1),dots,y_(m))rarr(y_(1),dots,y_(m),0,dots,0)\left(y_{1}, \ldots, y_{m}\right) \rightarrow\left(y_{1}, \ldots, y_{m}, 0, \ldots, 0\right)(y1,…,ym)→(y1,…,ym,0,…,0).
A famous theorem of Abhyankar-Moh and Suzuki proves that any epimorphism ϕ : k [ X , Y ] k [ T ] Ï• : k [ X , Y ] → k [ T ] phi:k[X,Y]rarr k[T]\phi: k[X, Y] \rightarrow k[T]Ï•:k[X,Y]→k[T] is rectifiable in characteristic zero [5,86]. On the other hand, in positive characteristic, there exist nonrectifiable epimorphisms from k [ X , Y ] k [ X , Y ] k[X,Y]k[X, Y]k[X,Y] to k [ T ] k [ T ] k[T]k[T]k[T] (see Segre [83], Nagata [71]). It is an open problem whether there exist nonrectifiable epimorphisms over the field of complex numbers (see [38]).
Asanuma has described an explicit method for constructing affine rings which are stably polynomial rings, by making use of nonrectifiable epimorphisms ([7], also see [38, PROPOSITION 3.7]). Such rings are considered to be potential candidates for counterexamples to the ZCP. For instance, when k k kkk is of positive characteristic, nonrectifiable epimorphisms from k [ X , Y ] k [ X , Y ] k[X,Y]k[X, Y]k[X,Y] to k [ T ] k [ T ] k[T]k[T]k[T] yield counterexamples to the ZCP.
Let ϕ : R [ X , Y , Z ] R [ T ] Ï• : R [ X , Y , Z ] → R [ T ] phi:R[X,Y,Z]rarrR[T]\phi: \mathbb{R}[X, Y, Z] \rightarrow \mathbb{R}[T]Ï•:R[X,Y,Z]→R[T] be defined by
ϕ ( X ) = T 3 3 T , ϕ ( Y ) = T 4 4 T 2 , ϕ ( Z ) = T 5 10 T Ï• ( X ) = T 3 − 3 T , Ï• ( Y ) = T 4 − 4 T 2 , Ï• ( Z ) = T 5 − 10 T phi(X)=T^(3)-3T,quad phi(Y)=T^(4)-4T^(2),quad phi(Z)=T^(5)-10 T\phi(X)=T^{3}-3 T, \quad \phi(Y)=T^{4}-4 T^{2}, \quad \phi(Z)=T^{5}-10 TÏ•(X)=T3−3T,Ï•(Y)=T4−4T2,Ï•(Z)=T5−10T
Shastri constructed the above epimorphism ϕ Ï• phi\phiÏ• and proved that it defines a nonrectifiable (polynomial) embedding of the trefoil knot in A R 3 A R 3 A_(R)^(3)\mathbb{A}_{\mathbb{R}}^{3}AR3 [84]. Using a result of Serre [63, tHEoREM 1, P. 281], one knows that ker ( ϕ ) = ( f , g ) ker ⁡ ( Ï• ) = ( f , g ) ker(phi)=(f,g)\operatorname{ker}(\phi)=(f, g)ker⁡(Ï•)=(f,g) for some f , g k [ X , Y , Z ] f , g ∈ k [ X , Y , Z ] f,g in k[X,Y,Z]f, g \in k[X, Y, Z]f,g∈k[X,Y,Z]. Using f f fff and g g ggg, Asanuma constructed the ring B = R [ T ] [ X , Y , Z , U , V ] / ( T d U f , T d V g ) B = R [ T ] [ X , Y , Z , U , V ] / T d U − f , T d V − g B=R[T][X,Y,Z,U,V]//(T^(d)U-f,T^(d)V-g)B=\mathbb{R}[T][X, Y, Z, U, V] /\left(T^{d} U-f, T^{d} V-g\right)B=R[T][X,Y,Z,U,V]/(TdU−f,TdV−g) and proved that B [ 1 ] = R [ T ] [ 4 ] = R [ 5 ] ( c f B [ 1 ] = R [ T ] [ 4 ] = R [ 5 ] ( c f B^([1])=R[T]^([4])=R^([5])(cfB^{[1]}=\mathbb{R}[T]^{[4]}=\mathbb{R}^{[5]}(\mathrm{cf}B[1]=R[T][4]=R[5](cf. [7, cORoLLARY 4.2]). He asked [7, REMARK 7.8]:
Question 2. Is B = R [ 4 ] B = R [ 4 ] B=R^([4])B=\mathbb{R}^{[4]}B=R[4] ?
The interesting aspect of the question is that once the problem gets solved, irrespective of whether the answer is "Yes" or "No," that is, either way, one would have solved a major problem in Affine Algebraic Geometry. Indeed:
If B = R [ 4 ] B = R [ 4 ] B=R^([4])B=\mathbb{R}^{[4]}B=R[4], then there exist nonlinearizable R R ∗ R^(**)\mathbb{R}^{*}R∗-actions on the affine four-space A R 4 A R 4 A_(R)^(4)\mathbb{A}_{\mathbb{R}}^{4}AR4.
If B R [ 4 ] B ≠ R [ 4 ] B!=R^([4])B \neq \mathbb{R}^{[4]}B≠R[4], then clearly B B BBB is a counterexample to the ZCP!!

3. CHARACTERIZATION PROBLEM

The Characterization Problem in affine algebraic geometry seeks a "useful characterization" of the polynomial ring or, equivalently (when the ground field is algebraically
closed), an affine n n nnn-space. For instance, the following two results give respectively an algebraic and a topological characterization of k [ 1 ] k [ 1 ] k^([1])k^{[1]}k[1] (or A C 1 A C 1 A_(C)^(1)\mathbb{A}_{\mathbb{C}}^{1}AC1 ).
Theorem 3.1. Let k k kkk be an algebraically closed field of characteristic zero. Then the polynomial ring k [ 1 ] k [ 1 ] k^([1])k^{[1]}k[1] is the only one-dimensional affine UFD with A = k A ∗ = k ∗ A^(**)=k^(**)A^{*}=k^{*}A∗=k∗.
Theorem 3.2. Let k k kkk be the field of complex numbers C C C\mathbb{C}C. Then the affine line A C 1 A C 1 A_(C)^(1)\mathbb{A}_{\mathbb{C}}^{1}AC1 is the only acyclic normal curve.
While the Characterization Problem is one of the most important problems in affine algebraic geometry in its own right, it is also closely related to some of the challenging open problems on the affine space like the "Cancellation Problem." For instance, each of the above characterizations of k [ 1 ] k [ 1 ] k^([1])k^{[1]}k[1] immediately solves the Cancellation Problem in dimension one: A [ 1 ] = k [ 2 ] A = k [ 1 ] A [ 1 ] = k [ 2 ] ⟹ A = k [ 1 ] A^([1])=k^([2])Longrightarrow A=k^([1])A^{[1]}=k^{[2]} \Longrightarrow A=k^{[1]}A[1]=k[2]⟹A=k[1]. The complexity of the Characterization Problem increases with the dimension of the rings.
In his attempt to solve the Cancellation Problem for the affine plane, Ramanujam obtained a remarkable topological characterization of the affine plane C 2 C 2 C^(2)\mathbb{C}^{2}C2 in 1971 [72]. He proved that
Theorem 3.3. C 2 C 2 C^(2)\mathbb{C}^{2}C2 is the only contractible smooth surface which is simply connected at infinity.
Ramanujam also constructed contractible surfaces which are not isomorphic to C 2 C 2 C^(2)\mathbb{C}^{2}C2. Soon, in 1975, Miyanishi [67] obtained an algebraic characterization of the polynomial ring k [ 2 ] k [ 2 ] k^([2])k^{[2]}k[2]. He proved that
Theorem 3.4. Let k k kkk be an algebraically closed field of characteristic zero and A A AAA be a twodimensional affine factorial domain over k k kkk. Then A = k [ 2 ] A = k [ 2 ] A=k^([2])A=k^{[2]}A=k[2] if and only if it satisfies the following:
(i) A = k A ∗ = k ∗ A^(**)=k^(**)A^{*}=k^{*}A∗=k∗.
(ii) There exists an element f A f ∈ A f in Af \in Af∈A and a subring B B BBB of A A AAA such that A [ f 1 ] = A f − 1 = A[f^(-1)]=A\left[f^{-1}\right]=A[f−1]= B [ f 1 ] [ 1 ] B f − 1 [ 1 ] B[f^(-1)]^([1])B\left[f^{-1}\right]^{[1]}B[f−1][1].
This algebraic characterization was used by Fujita, Miyanishi, and Sugie [43,70] to solve the Cancellation Problem for k [ X , Y ] k [ X , Y ] k[X,Y]k[X, Y]k[X,Y]. In 2002 [50], using methods of Mumford and Ramanujam, Gurjar gave a topological proof of the cancellation property of C [ X , Y ] C [ X , Y ] C[X,Y]\mathbb{C}[X, Y]C[X,Y].
Remarkable characterizations of the affine three space were obtained by Miyanishi [68] and Kaliman [56] (also see [69] for a beautiful survey). We state below the version of Kaliman.
Theorem 3.5. Let A be a three-dimensional smooth factorial affine domain over the field of complex numbers C C C\mathbb{C}C. Let X = Spec A X = Spec ⁡ A X=Spec AX=\operatorname{Spec} AX=Spec⁡A. Then A = C [ 3 ] A = C [ 3 ] A=C^([3])A=\mathbb{C}^{[3]}A=C[3] if and only if it satisfies the following:
(i) A = C A ∗ = C ∗ quadA^(**)=C^(**)\quad A^{*}=\mathbb{C}^{*}A∗=C∗.
(ii) H 3 ( X , Z ) = 0 H 3 ( X , Z ) = 0 H_(3)(X,Z)=0H_{3}(X, \mathbb{Z})=0H3(X,Z)=0, or X X XXX is contractible.
(iii) X X XXX contains a cylinder-like open set V V VVV such that V U × A 2 V ≅ U × A 2 V~=U xxA^(2)V \cong U \times \mathbb{A}^{2}V≅U×A2 for some curve U U UUU and each irreducible component of the complement X V X ∖ V X\\VX \backslash VX∖V has at most isolated singularities.
When A [ 1 ] = C [ 4 ] A [ 1 ] = C [ 4 ] A^([1])=C^([4])A^{[1]}=\mathbb{C}^{[4]}A[1]=C[4], it is easy to see that A A AAA possesses properties (i) and (ii) of Theorem 3.5. Thus, by Theorem 3.5 , the ZCP for C [ 3 ] C [ 3 ] C^([3])\mathbb{C}^{[3]}C[3] reduces to examining whether condition (iii) necessarily holds for a C C C\mathbb{C}C-algebra A A AAA satisfying A [ 1 ] = C [ 4 ] A [ 1 ] = C [ 4 ] A^([1])=C^([4])A^{[1]}=\mathbb{C}^{[4]}A[1]=C[4].
In [29], we have obtained another characterization of the affine three-space using certain invariants of an affine domain defined by locally nilpotent derivations. We state it below.
Locally nilpotent derivations and a characterization of C [ 3 ] C [ 3 ] C^([3])\mathbb{C}^{[3]}C[3]. Let B B BBB be an affine domain over a field k k kkk of characteristic zero. A k k kkk-linear derivation D D DDD on B B BBB is said to be a locally nilpotent derivation if, for any a B a ∈ B a in Ba \in Ba∈B there exists an integer n n nnn (depending on a a aaa ) satisfying D n ( a ) = 0 D n ( a ) = 0 D^(n)(a)=0D^{n}(a)=0Dn(a)=0. Let LND ( B ) LND ⁡ ( B ) LND(B)\operatorname{LND}(B)LND⁡(B) denote the set of all locally nilpotent k k kkk-derivations of B B BBB and let
LND ( B ) = { D LND ( B ) D s = 1 for some s B } LND ∗ ⁡ ( B ) = { D ∈ LND ⁡ ( B ) ∣ D s = 1  for some  s ∈ B } LND^(**)(B)={D in LND(B)∣Ds=1" for some "s in B}\operatorname{LND}^{*}(B)=\{D \in \operatorname{LND}(B) \mid D s=1 \text { for some } s \in B\}LND∗⁡(B)={D∈LND⁡(B)∣Ds=1 for some s∈B}
Then we define
ML ( B ) := D LND ( B ) ker D and ML ( B ) := D LND ( B ) ker D ML ⁡ ( B ) := ⋂ D ∈ LND ⁡ ( B )   ker ⁡ D  and  ML ∗ ⁡ ( B ) := ⋂ D ∈ LND ∗ ⁡ ( B )   ker ⁡ D ML(B):=nnn_(D in LND(B))ker D quad" and "quadML^(**)(B):=nnn_(D inLND^(**)(B))ker D\operatorname{ML}(B):=\bigcap_{D \in \operatorname{LND}(B)} \operatorname{ker} D \quad \text { and } \quad \operatorname{ML}^{*}(B):=\bigcap_{D \in \operatorname{LND}^{*}(B)} \operatorname{ker} DML⁡(B):=⋂D∈LND⁡(B)ker⁡D and ML∗⁡(B):=⋂D∈LND∗⁡(B)ker⁡D
The above ML ( B ) ML ⁡ ( B ) ML(B)\operatorname{ML}(B)ML⁡(B), introduced by Makar-Limanov, is now called the Makar-Limanov invariant of B B BBB; ML* ( B ) ∗ ( B ) ^(**)(B){ }^{*}(B)∗(B) was introduced by Freudenburg in [41, P. 237]. We call it the MakarLimanov-Freudenburg invariant or ML-F invariant. If LND ( B ) = LND ∗ ⁡ ( B ) = ∅ LND^(**)(B)=O/\operatorname{LND}^{*}(B)=\emptysetLND∗⁡(B)=∅, we define M L ( B ) M L ∗ ( B ) ML^(**)(B)\mathrm{ML}^{*}(B)ML∗(B) to be B B BBB. We have obtained the following theorem [29, THEOREM 4.6].
Theorem 3.6. Let A be a three-dimensional affine factorial domain over an algebraically closed field k k kkk of characteristic zero. Then the following are equivalent:
(I) A = k [ 3 ] A = k [ 3 ] A=k^([3])A=k^{[3]}A=k[3].
(II) ML ( A ) = k ML ∗ ⁡ ( A ) = k ML^(**)(A)=k\operatorname{ML}^{*}(A)=kML∗⁡(A)=k.
(III) ML ( A ) = k ML ⁡ ( A ) = k ML(A)=k\operatorname{ML}(A)=kML⁡(A)=k and ML ( A ) A ML ∗ ⁡ ( A ) ≠ A ML^(**)(A)!=A\operatorname{ML}^{*}(A) \neq AML∗⁡(A)≠A.
A similar result has also been proved in dimension two under weaker hypotheses [29, theorem 3.8]. The above characterization of the affine three-space does not extend to higher dimensions [29, EXAMPLE 5.6]. So far, no suitable characterization of the affine n n nnn-space for n 4 n ≥ 4 n >= 4n \geq 4n≥4 is known to the author.

4. AFFINE FIBRATIONS

Let R R RRR be a commutative ring. A fundamental theorem of Bass-Connell-Wright and Suslin [ 10 , 85 ] [ 10 , 85 ] [10,85][10,85][10,85] on the structure of locally polynomial algebras states that:
Theorem 4.1. Let A A AAA be a finitely presented algebra over a ring R. Suppose that for each maximal ideal m m mmm of R , A m = R m [ n ] R , A m = R m [ n ] R,A_(m)=R_(m)^([n])R, A_{m}=R_{m}^{[n]}R,Am=Rm[n] for some integer n 0 n ≥ 0 n >= 0n \geq 0n≥0. Then A Sym R ( P ) A ≅ Sym R ⁡ ( P ) A~=Sym_(R)(P)A \cong \operatorname{Sym}_{R}(P)A≅SymR⁡(P) for some finitely generated projective R R RRR-module P P PPP of rank n n nnn.
Now for a prime ideal P P PPP of R R RRR, let k ( P ) k ( P ) k(P)k(P)k(P) denote the residue field R P / P R P R P / P R P R_(P)//PR_(P)R_{P} / P R_{P}RP/PRP. The area of affine fibrations seeks to derive information about the structure and properties of an R R RRR-algebra A A AAA from the information about the fiber rings A R k ( P ) ( = A P / P A P ) A ⊗ R k ( P ) = A P / P A P Aox_(R)k(P)(=A_(P)//PA_(P))A \otimes_{R} k(P)\left(=A_{P} / P A_{P}\right)A⊗Rk(P)(=AP/PAP) of A A AAA at the points P P PPP of the prime spectrum of R R RRR, i.e., at the prime ideals P P PPP of R R RRR.
An R R RRR-algebra A A AAA is said to be an A n A n A^(n)\mathbb{A}^{n}An-fibration over R R RRR if A A AAA is a finitely generated flat R R RRR-algebra and for each prime ideal P P PPP of R , A R k ( P ) = k ( P ) [ n ] R , A ⊗ R k ( P ) = k ( P ) [ n ] R,Aox_(R)k(P)=k(P)^([n])R, A \otimes_{R} k(P)=k(P)^{[n]}R,A⊗Rk(P)=k(P)[n].
The most important problem on A n A n A^(n)\mathbb{A}^{n}An-fibrations, due to Veǐsfeǐler and Dolgačev [87], can be formulated as follows:
Question 3. Let R R RRR be a Noetherian domain of dimension d d ddd and A A AAA be an A n A n A^(n)\mathbb{A}^{n}An-fibration over R R RRR.
(i) If R R RRR is regular, is A Sym R ( Q ) A ≅ Sym R ⁡ ( Q ) A~=Sym_(R)(Q)A \cong \operatorname{Sym}_{R}(Q)A≅SymR⁡(Q) for some projective module Q Q QQQ over R R RRR ? (In particular, if R R RRR is regular local, is then A = R [ n ] A = R [ n ] A=R^([n])A=R^{[n]}A=R[n] ?)
(ii) In general, what can one say about the structure of A A AAA ?
Question 3 is considered a hard problem. When n = 1 n = 1 n=1n=1n=1, it has an affirmative answer for all d d ddd. This has been established in the works of Kambayashi, Miyanishi, and Wright [59,60]. Their results were further refined by Dutta who showed that it is enough to assume the fiber conditions only on generic and codimension-one fibers ([34]; also see [14, 17, 40]).
In case n = 2 , d = 1 n = 2 , d = 1 n=2,d=1n=2, d=1n=2,d=1, and R R RRR contains the field of rational numbers, an important theorem of Sathaye [81] gives an affirmative answer to Question 3 (i). To prove this theorem, Sathaye first generalized the Abhyankar-Moh expansion techniques originally developed over k [ [ x ] ] k [ [ x ] ] k[[x]]k[[x]]k[[x]] to k [ [ x 1 , , x n ] ] k x 1 , … , x n k[[x_(1),dots,x_(n)]]k\left[\left[x_{1}, \ldots, x_{n}\right]\right]k[[x1,…,xn]] [80]. The expansion techniques were used by Abhyankar-Moh to prove their famous epimorphism theorem. The generalized expansion techniques were further developed by Sathaye [82] to prove a conjecture of Daigle and Freudenburg. The result was a crucial step in Daigle-Freudenburg's theorem that the kernel of any triangular derivation of k [ X 1 , X 2 , X 3 , X 4 ] k X 1 , X 2 , X 3 , X 4 k[X_(1),X_(2),X_(3),X_(4)]k\left[X_{1}, X_{2}, X_{3}, X_{4}\right]k[X1,X2,X3,X4] is a finitely generated k k kkk-algebra [23].
When the residue field of R R RRR is of positive characteristic, Asanuma has shown in [6, THEOREM 5.1] that Question 3 (i) has a negative answer for n = 2 , d = 1 n = 2 , d = 1 n=2,d=1n=2, d=1n=2,d=1, and the author has generalized Asanuma's ring [47] to give a negative answer to Question 3 (i) for n = 2 n = 2 n=2n=2n=2 and any d > 1 d > 1 d > 1d>1d>1 (also see [48]). In Theorem 5.4, the author proved that in a special situation A 2 A 2 A^(2)\mathbb{A}^{2}A2-fibration is indeed trivial.
However, if n = 2 , d = 2 n = 2 , d = 2 n=2,d=2n=2, d=2n=2,d=2, and R R RRR contains the field of rational numbers, Question 3 (i) is an open problem. A candidate counterexample is discussed in Section 7.
In the context of Question 3 (ii), a deep work of Asanuma [6] provides a stable structure theorem for A A AAA. As a consequence of Asanuma's structure theorem, it follows that if R R RRR is regular local, then there exists an integer m 0 m ≥ 0 m >= 0m \geq 0m≥0 such that A [ m ] = R [ m + n ] A [ m ] = R [ m + n ] A^([m])=R^([m+n])A^{[m]}=R^{[m+n]}A[m]=R[m+n]. Thus it is very tempting to look for possible counterexamples to the affine fibration problem in order to
obtain possible counterexamples to the ZCP in characteristic zero. One can see [ 12 , 24 , 36 , 37 ] [ 12 , 24 , 36 , 37 ] [12,24,36,37][12,24,36,37][12,24,36,37] and [38, SECTION 3.1] for more results on affine fibrations.
So far we have considered affine fibrations where the fibre rings are polynomial rings. Bhatwadekar and Dutta have obtained some nice results on rings whose fiber rings are of the form k [ X , 1 / X ] [ 15 , 16 ] k [ X , 1 / X ] [ 15 , 16 ] k[X,1//X][15,16]k[X, 1 / X][15,16]k[X,1/X][15,16]. Later Bhatwadekar, the author, and A. Abhyankar studied rings whose fiber rings are Laurent polynomial algebras or rings of the form k [ X , 1 / f ( X ) ] k [ X , 1 / f ( X ) ] k[X,1//f(X)]k[X, 1 / f(X)]k[X,1/f(X)], or of the form k [ X , Y , 1 / ( a X + b ) , 1 / ( c Y + d ) ] k [ X , Y , 1 / ( a X + b ) , 1 / ( c Y + d ) ] k[X,Y,1//(aX+b),1//(cY+d)]k[X, Y, 1 /(a X+b), 1 /(c Y+d)]k[X,Y,1/(aX+b),1/(cY+d)] for some a , b , c , d k [ 1 , 2 , 18 , 19 , 44 ] a , b , c , d ∈ k [ 1 , 2 , 18 , 19 , 44 ] a,b,c,d in k[1,2,18,19,44]a, b, c, d \in k[1,2,18,19,44]a,b,c,d∈k[1,2,18,19,44]. One of the results of Bhatwadekar and the author provides a Laurent polynomial analogue of Theorem 4.1 and the affine fibration problem Question 3. More generally, we have [19, THEOREMS A AND C]:
Theorem 4.2. Let R R RRR be a Noetherian normal domain with field of fractions K K KKK and A A AAA be a faithfully flat R R RR